2. Radicals and Pythagorean Theorem
Solutions of equations obtained by the factoring–method
are precise - they are whole numbers or fractions.
3. Radicals and Pythagorean Theorem
Solutions of equations obtained by the factoring–method
are precise - they are whole numbers or fractions.
Now let’s look at numbers that can’t be easily computed.
We need a scientific calculator to estimate these numbers.
4. “9 is the square of 3” may be rephrased backwards as
“3 is the square root of 9”.
Radicals and Pythagorean Theorem
Square Root
Solutions of equations obtained by the factoring–method
are precise - they are whole numbers or fractions.
Now let’s look at numbers that can’t be easily computed.
We need a scientific calculator to estimate these numbers.
5. “9 is the square of 3” may be rephrased backwards as
“3 is the square root of 9”.
Radicals and Pythagorean Theorem
Definition: If a is > 0, and a2 = x, then we say a is the square
root of x. This is written as sqrt(x) = a, or x = a.
Square Root
Solutions of equations obtained by the factoring–method
are precise - they are whole numbers or fractions.
Now let’s look at numbers that can’t be easily computed.
We need a scientific calculator to estimate these numbers.
6. “9 is the square of 3” may be rephrased backwards as
“3 is the square root of 9”.
Example A.
a. Sqrt(16) =
c.–3 =
Radicals and Pythagorean Theorem
Definition: If a is > 0, and a2 = x, then we say a is the square
root of x. This is written as sqrt(x) = a, or x = a.
b. 1/9 =
d. 3 =
Square Root
Solutions of equations obtained by the factoring–method
are precise - they are whole numbers or fractions.
Now let’s look at numbers that can’t be easily computed.
We need a scientific calculator to estimate these numbers.
7. “9 is the square of 3” may be rephrased backwards as
“3 is the square root of 9”.
Example A.
a. Sqrt(16) = 4
c.–3 =
Radicals and Pythagorean Theorem
Definition: If a is > 0, and a2 = x, then we say a is the square
root of x. This is written as sqrt(x) = a, or x = a.
b. 1/9 =
d. 3 =
Square Root
Solutions of equations obtained by the factoring–method
are precise - they are whole numbers or fractions.
Now let’s look at numbers that can’t be easily computed.
We need a scientific calculator to estimate these numbers.
8. “9 is the square of 3” may be rephrased backwards as
“3 is the square root of 9”.
Example A.
a. Sqrt(16) = 4
c.–3 =
Radicals and Pythagorean Theorem
Definition: If a is > 0, and a2 = x, then we say a is the square
root of x. This is written as sqrt(x) = a, or x = a.
b. 1/9 = 1/3
d. 3 =
Square Root
Solutions of equations obtained by the factoring–method
are precise - they are whole numbers or fractions.
Now let’s look at numbers that can’t be easily computed.
We need a scientific calculator to estimate these numbers.
9. “9 is the square of 3” may be rephrased backwards as
“3 is the square root of 9”.
Example A.
a. Sqrt(16) = 4
c.–3 = doesn’t exist
Radicals and Pythagorean Theorem
Definition: If a is > 0, and a2 = x, then we say a is the square
root of x. This is written as sqrt(x) = a, or x = a.
b. 1/9 = 1/3
d. 3 =
Square Root
Solutions of equations obtained by the factoring–method
are precise - they are whole numbers or fractions.
Now let’s look at numbers that can’t be easily computed.
We need a scientific calculator to estimate these numbers.
10. “9 is the square of 3” may be rephrased backwards as
“3 is the square root of 9”.
Example A.
a. Sqrt(16) = 4
c.–3 = doesn’t exist
Radicals and Pythagorean Theorem
Definition: If a is > 0, and a2 = x, then we say a is the square
root of x. This is written as sqrt(x) = a, or x = a.
b. 1/9 = 1/3
d. 3 = 1.732.. (calculator)
Square Root
Solutions of equations obtained by the factoring–method
are precise - they are whole numbers or fractions.
Now let’s look at numbers that can’t be easily computed.
We need a scientific calculator to estimate these numbers.
11. “9 is the square of 3” may be rephrased backwards as
“3 is the square root of 9”.
Example A.
a. Sqrt(16) = 4
c.–3 = doesn’t exist
Radicals and Pythagorean Theorem
Definition: If a is > 0, and a2 = x, then we say a is the square
root of x. This is written as sqrt(x) = a, or x = a.
b. 1/9 = 1/3
d. 3 = 1.732.. (calculator)
Note that the square of both +3 and –3 is 9, but we
designate sqrt(9) or 9 to be +3.
Square Root
Solutions of equations obtained by the factoring–method
are precise - they are whole numbers or fractions.
Now let’s look at numbers that can’t be easily computed.
We need a scientific calculator to estimate these numbers.
12. “9 is the square of 3” may be rephrased backwards as
“3 is the square root of 9”.
Example A.
a. Sqrt(16) = 4
c.–3 = doesn’t exist
Radicals and Pythagorean Theorem
Definition: If a is > 0, and a2 = x, then we say a is the square
root of x. This is written as sqrt(x) = a, or x = a.
b. 1/9 = 1/3
d. 3 = 1.732.. (calculator)
Note that the square of both +3 and –3 is 9, but we
designate sqrt(9) or 9 to be +3. We say “–3” is the
“negative of the square root of 9”.
Solutions of equations obtained by the factoring–method
are precise - they are whole numbers or fractions.
Now let’s look at numbers that can’t be easily computed.
We need a scientific calculator to estimate these numbers.
Square Root
14. 0 02 = 0 0 = 0
1 12 = 1 1 = 1
2 22 = 4 4 = 2
3 32 = 9 9 = 3
4 42 = 16 16 = 4
5 52 = 25 25 = 5
6 62 = 36 36 = 6
7 72 = 49 49 = 7
8 82 = 64 64 = 8
9 92 = 81 81 = 9
10 102 = 100 100 = 10
11 112 = 121 121 = 11
Radicals and Pythagorean Theorem
Following are the square numbers and square-roots that one
needs to memorize. These numbers are special because
many mathematics exercises utilize square numbers.
15. 0 02 = 0 0 = 0
1 12 = 1 1 = 1
2 22 = 4 4 = 2
3 32 = 9 9 = 3
4 42 = 16 16 = 4
5 52 = 25 25 = 5
6 62 = 36 36 = 6
7 72 = 49 49 = 7
8 82 = 64 64 = 8
9 92 = 81 81 = 9
10 102 = 100 100 = 10
11 112 = 121 121 = 11
We may estimate the sqrt
of other small numbers using
this table.
Radicals and Pythagorean Theorem
Following are the square numbers and square-roots that one
needs to memorize. These numbers are special because
many mathematics exercises utilize square numbers.
16. 0 02 = 0 0 = 0
1 12 = 1 1 = 1
2 22 = 4 4 = 2
3 32 = 9 9 = 3
4 42 = 16 16 = 4
5 52 = 25 25 = 5
6 62 = 36 36 = 6
7 72 = 49 49 = 7
8 82 = 64 64 = 8
9 92 = 81 81 = 9
10 102 = 100 100 = 10
11 112 = 121 121 = 11
We may estimate the sqrt
of other small numbers using
this table. For example,
25 < 30 < 36
Radicals and Pythagorean Theorem
Following are the square numbers and square-roots that one
needs to memorize. These numbers are special because
many mathematics exercises utilize square numbers.
17. 0 02 = 0 0 = 0
1 12 = 1 1 = 1
2 22 = 4 4 = 2
3 32 = 9 9 = 3
4 42 = 16 16 = 4
5 52 = 25 25 = 5
6 62 = 36 36 = 6
7 72 = 49 49 = 7
8 82 = 64 64 = 8
9 92 = 81 81 = 9
10 102 = 100 100 = 10
11 112 = 121 121 = 11
We may estimate the sqrt
of other small numbers using
this table. For example,
25 < 30 < 36
hence
25 < 30 <36
Radicals and Pythagorean Theorem
Following are the square numbers and square-roots that one
needs to memorize. These numbers are special because
many mathematics exercises utilize square numbers.
18. 0 02 = 0 0 = 0
1 12 = 1 1 = 1
2 22 = 4 4 = 2
3 32 = 9 9 = 3
4 42 = 16 16 = 4
5 52 = 25 25 = 5
6 62 = 36 36 = 6
7 72 = 49 49 = 7
8 82 = 64 64 = 8
9 92 = 81 81 = 9
10 102 = 100 100 = 10
11 112 = 121 121 = 11
We may estimate the sqrt
of other small numbers using
this table. For example,
25 < 30 < 36
hence
25 < 30 <36
or 5 < 30 < 6
Radicals and Pythagorean Theorem
Following are the square numbers and square-roots that one
needs to memorize. These numbers are special because
many mathematics exercises utilize square numbers.
19. 0 02 = 0 0 = 0
1 12 = 1 1 = 1
2 22 = 4 4 = 2
3 32 = 9 9 = 3
4 42 = 16 16 = 4
5 52 = 25 25 = 5
6 62 = 36 36 = 6
7 72 = 49 49 = 7
8 82 = 64 64 = 8
9 92 = 81 81 = 9
10 102 = 100 100 = 10
11 112 = 121 121 = 11
We may estimate the sqrt
of other small numbers using
this table. For example,
25 < 30 < 36
hence
25 < 30 <36
or 5 < 30 < 6
Since 30 is about half way
between 25 and 36,
Radicals and Pythagorean Theorem
Following are the square numbers and square-roots that one
needs to memorize. These numbers are special because
many mathematics exercises utilize square numbers.
20. 0 02 = 0 0 = 0
1 12 = 1 1 = 1
2 22 = 4 4 = 2
3 32 = 9 9 = 3
4 42 = 16 16 = 4
5 52 = 25 25 = 5
6 62 = 36 36 = 6
7 72 = 49 49 = 7
8 82 = 64 64 = 8
9 92 = 81 81 = 9
10 102 = 100 100 = 10
11 112 = 121 121 = 11
We may estimate the sqrt
of other small numbers using
this table. For example,
25 < 30 < 36
hence
25 < 30 <36
or 5 < 30 < 6
Since 30 is about half way
between 25 and 36,
so we estimate that30 5.5.
Radicals and Pythagorean Theorem
Following are the square numbers and square-roots that one
needs to memorize. These numbers are special because
many mathematics exercises utilize square numbers.
21. 0 02 = 0 0 = 0
1 12 = 1 1 = 1
2 22 = 4 4 = 2
3 32 = 9 9 = 3
4 42 = 16 16 = 4
5 52 = 25 25 = 5
6 62 = 36 36 = 6
7 72 = 49 49 = 7
8 82 = 64 64 = 8
9 92 = 81 81 = 9
10 102 = 100 100 = 10
11 112 = 121 121 = 11
We may estimate the sqrt
of other small numbers using
this table. For example,
25 < 30 < 36
hence
25 < 30 <36
or 5 < 30 < 6
Since 30 is about half way
between 25 and 36,
so we estimate that30 5.5.
In fact 30 5.47722….
Radicals and Pythagorean Theorem
Following are the square numbers and square-roots that one
needs to memorize. These numbers are special because
many mathematics exercises utilize square numbers.
22. 0 02 = 0 0 = 0
1 12 = 1 1 = 1
2 22 = 4 4 = 2
3 32 = 9 9 = 3
4 42 = 16 16 = 4
5 52 = 25 25 = 5
6 62 = 36 36 = 6
7 72 = 49 49 = 7
8 82 = 64 64 = 8
9 92 = 81 81 = 9
10 102 = 100 100 = 10
11 112 = 121 121 = 11
We may estimate the sqrt
of other small numbers using
this table. For example,
25 < 30 < 36
hence
25 < 30 <36
or 5 < 30 < 6
Since 30 is about half way
between 25 and 36,
so we estimate that30 5.5.
In fact 30 5.47722….
Radicals and Pythagorean Theorem
Following are the square numbers and square-roots that one
needs to memorize. These numbers are special because
many mathematics exercises utilize square numbers.
2 is especially important due
to the Pythagorean Theorem.
24. A right triangle is a triangle with one of its angles a right angle.
Radicals and Pythagorean Theorem
Pythagorean Theorem
a right angle
A right triangle
25. A right triangle is a triangle with one of its angles a right angle.
The following is the standard labeling of a right triangle.
Radicals and Pythagorean Theorem
Pythagorean Theorem
a right angle
A right triangle
26. A right triangle is a triangle with one of its angles a right angle.
The following is the standard labeling of a right triangle.
The longest side C, opposite from the right angle of a right
triangle is called the hypotenuse.
Radicals and Pythagorean Theorem
hypotenuse
C
Pythagorean Theorem
a right angle
A right triangle
27. A right triangle is a triangle with one of its angles a right angle.
The following is the standard labeling of a right triangle.
The longest side C, opposite from the right angle of a right
triangle is called the hypotenuse.
The shorter sides A and B are called the legs.
Radicals and Pythagorean Theorem
hypotenuse
legs
A
B
C
Pythagorean Theorem
a right angle
A right triangle
28. A right triangle is a triangle with one of its angles a right angle.
The following is the standard labeling of a right triangle.
The longest side C, opposite from the right angle of a right
triangle is called the hypotenuse.
The shorter sides A and B are called the legs.
Radicals and Pythagorean Theorem
hypotenuse
legs
A
B
C
Pythagorean Theorem
Given a right triangle with labeling as shown,
then A2 + B2 = C2
a right angle
A right triangle
29. Radicals and Pythagorean Theorem
There are two types of problems that use the Pythagorean
Theorem to find the sides of the right triangles
30. Radicals and Pythagorean Theorem
There are two types of problems that use the Pythagorean
Theorem to find the sides of the right triangles-finding the
hypotenuse versus finding the legs.
31. Example B. Find the missing side of the following right
triangles. Find the exact answer and the approximate answer.
Draw.
a. a = 5, b = 12, c = ?
Radicals and Pythagorean Theorem
There are two types of problems that use the Pythagorean
Theorem to find the sides of the right triangles-finding the
hypotenuse versus finding the legs.
32. Example B. Find the missing side of the following right
triangles. Find the exact answer and the approximate answer.
Draw.
a. a = 5, b = 12, c = ?
Radicals and Pythagorean Theorem
There are two types of problems that use the Pythagorean
Theorem to find the sides of the right triangles-finding the
hypotenuse versus finding the legs.
33. Example B. Find the missing side of the following right
triangles. Find the exact answer and the approximate answer.
Draw.
a. a = 5, b = 12, c = ?
Since it is a right triangle,
122 + 52 = c2
Radicals and Pythagorean Theorem
There are two types of problems that use the Pythagorean
Theorem to find the sides of the right triangles-finding the
hypotenuse versus finding the legs.
34. Example B. Find the missing side of the following right
triangles. Find the exact answer and the approximate answer.
Draw.
a. a = 5, b = 12, c = ?
Since it is a right triangle,
122 + 52 = c2
144 + 25 = c2
Radicals and Pythagorean Theorem
There are two types of problems that use the Pythagorean
Theorem to find the sides of the right triangles-finding the
hypotenuse versus finding the legs.
35. Example B. Find the missing side of the following right
triangles. Find the exact answer and the approximate answer.
Draw.
a. a = 5, b = 12, c = ?
Since it is a right triangle,
122 + 52 = c2
144 + 25 = c2
169 = c2
Radicals and Pythagorean Theorem
There are two types of problems that use the Pythagorean
Theorem to find the sides of the right triangles-finding the
hypotenuse versus finding the legs.
36. Example B. Find the missing side of the following right
triangles. Find the exact answer and the approximate answer.
Draw.
a. a = 5, b = 12, c = ?
Since it is a right triangle,
122 + 52 = c2
144 + 25 = c2
169 = c2
So c = ±169 = ±13
Radicals and Pythagorean Theorem
There are two types of problems that use the Pythagorean
Theorem to find the sides of the right triangles-finding the
hypotenuse versus finding the legs.
37. Example B. Find the missing side of the following right
triangles. Find the exact answer and the approximate answer.
Draw.
a. a = 5, b = 12, c = ?
Since it is a right triangle,
122 + 52 = c2
144 + 25 = c2
169 = c2
So c = ±169 = ±13
Since length can’t be
negative, therefore c = 13.
Radicals and Pythagorean Theorem
There are two types of problems that use the Pythagorean
Theorem to find the sides of the right triangles-finding the
hypotenuse versus finding the legs.
38. b. a = 5, c = 12, b = ?
Radicals and Pythagorean Theorem
39. b. a = 5, c = 12, b = ?
Radicals and Pythagorean Theorem
40. b. a = 5, c = 12, b = ?
Since it is a right triangle,
b2 + 52 = 122
Radicals and Pythagorean Theorem
41. b. a = 5, c = 12, b = ?
Since it is a right triangle,
b2 + 52 = 122
b2 + 25 = 144
Radicals and Pythagorean Theorem
42. b. a = 5, c = 12, b = ?
Since it is a right triangle,
b2 + 52 = 122
b2 + 25 = 144
b2 = 144 – 25
Radicals and Pythagorean Theorem
43. b. a = 5, c = 12, b = ?
Since it is a right triangle,
b2 + 52 = 122
b2 + 25 = 144
b2 = 144 – 25
So b = ±119 ±10.9.
Radicals and Pythagorean Theorem
44. b. a = 5, c = 12, b = ?
Since it is a right triangle,
b2 + 52 = 122
b2 + 25 = 144
b2 = 144 – 25
So b = ±119 ±10.9.
But length can’t be negative,
therefore b = 119 10.9
Radicals and Pythagorean Theorem
45. Equations of the form x2 = c has two answers:
x = +c or –c if c>0.
Radicals and Pythagorean Theorem
46. Equations of the form x2 = c has two answers:
x = +c or –c if c>0.
Radicals and Pythagorean Theorem
Example C. Solve the following equations.
a. x2 = 25
47. Equations of the form x2 = c has two answers:
x = +c or –c if c>0.
Radicals and Pythagorean Theorem
Example C. Solve the following equations.
a. x2 = 25
x = ±25
48. Equations of the form x2 = c has two answers:
x = +c or –c if c>0.
Radicals and Pythagorean Theorem
Example C. Solve the following equations.
a. x2 = 25
x = ±25 = ±5
49. Equations of the form x2 = c has two answers:
x = +c or –c if c>0.
Radicals and Pythagorean Theorem
Example C. Solve the following equations.
a. x2 = 25
x = ±25 = ±5
b. x2 = –4
50. Equations of the form x2 = c has two answers:
x = +c or –c if c>0.
Radicals and Pythagorean Theorem
Example C. Solve the following equations.
a. x2 = 25
x = ±25 = ±5
b. x2 = –4
Solution does not exist.
51. Equations of the form x2 = c has two answers:
x = +c or –c if c>0.
Radicals and Pythagorean Theorem
Example C. Solve the following equations.
a. x2 = 25
x = ±25 = ±5
b. x2 = –4
Solution does not exist.
c. x2 = 8
52. Equations of the form x2 = c has two answers:
x = +c or –c if c>0.
Radicals and Pythagorean Theorem
Example C. Solve the following equations.
a. x2 = 25
x = ±25 = ±5
b. x2 = –4
Solution does not exist.
c. x2 = 8
x = ±8
53. Equations of the form x2 = c has two answers:
x = +c or –c if c>0.
Radicals and Pythagorean Theorem
Example C. Solve the following equations.
a. x2 = 25
x = ±25 = ±5
b. x2 = –4
Solution does not exist.
c. x2 = 8
x = ±8 ±2.8284.. by calculator
54. Equations of the form x2 = c has two answers:
x = +c or –c if c>0.
Radicals and Pythagorean Theorem
Example C. Solve the following equations.
a. x2 = 25
x = ±25 = ±5
b. x2 = –4
Solution does not exist.
c. x2 = 8
x = ±8 ±2.8284.. by calculator
exact answer approximate answer
55. Equations of the form x2 = c has two answers:
x = +c or –c if c>0.
Radicals and Pythagorean Theorem
Example C. Solve the following equations.
a. x2 = 25
x = ±25 = ±5
b. x2 = –4
Solution does not exist.
c. x2 = 8
x = ±8 ±2.8284.. by calculator
exact answer approximate answer
In geometry square roots show up in distance problems
because of Pythagorean theorem.
57. Radicals and Pythagorean Theorem
3
Higher Root
If r3 = x, then we say a is the cube root of x and we write this
as x = r.
58. Radicals and Pythagorean Theorem
3
Example E.
a. 8 = 2 b. –1 = –1 c.–27 = –3
3 3 3
Higher Root
If r3 = x, then we say a is the cube root of x and we write this
as x = r.
59. Radicals and Pythagorean Theorem
3
Example E.
a. 8 = 2 b. –1 = –1 c.–27 = –3
3 3 3
Higher Root
If r3 = x, then we say a is the cube root of x and we write this
as x = r.
60. Radicals and Pythagorean Theorem
3
Example E.
a. 8 = 2 b. –1 = –1 c.–27 = –3
3 3 3
Higher Root
If r3 = x, then we say a is the cube root of x and we write this
as x = r.
61. Radicals and Pythagorean Theorem
3
Example E.
a. 8 = 2 b. –1 = –1 c.–27 = –3
3 3 3
Higher Root
If r3 = x, then we say a is the cube root of x and we write this
as x = r.
62. Radicals and Pythagorean Theorem
3
k
Example E.
a. 8 = 2 b. –1 = –1 c.–27 = –3
3 3 3
Higher Root
If r3 = x, then we say a is the cube root of x and we write this
as x = r. In general, if r k = x then we say r is the k’th root of x,
and we write it as a = x.
63. Radicals and Pythagorean Theorem
3
k
k
Example E.
a. 8 = 2 b. –1 = –1 c.–27 = –3
3 3 3
Higher Root
If r3 = x, then we say a is the cube root of x and we write this
as x = r. In general, if r k = x then we say r is the k’th root of x,
and we write it as a = x. In the cases of even roots,
i.e. k = 2, 4, 6, … we must have x > 0 so that x = a > 0.
64. Radicals and Pythagorean Theorem
3
k
k
Example E.
a. 8 = 2 b. –1 = –1 c.–27 = –3
d. 16 = 2 e. –16 = not real f. 10 ≈ 2.15..
3 3 3
Higher Root
If r3 = x, then we say a is the cube root of x and we write this
as x = r. In general, if r k = x then we say r is the k’th root of x,
and we write it as a = x. In the cases of even roots,
i.e. k = 2, 4, 6, … we must have x > 0 so that x = a > 0.
4 4 3
65. Radicals and Pythagorean Theorem
3
k
k
Example E.
a. 8 = 2 b. –1 = –1 c.–27 = –3
d. 16 = 2 e. –16 = not real f. 10 ≈ 2.15..
3 3 3
Higher Root
If r3 = x, then we say a is the cube root of x and we write this
as x = r. In general, if r k = x then we say r is the k’th root of x,
and we write it as a = x. In the cases of even roots,
i.e. k = 2, 4, 6, … we must have x > 0 so that x = a > 0.
4 4 3
66. Radicals and Pythagorean Theorem
3
k
k
Example E.
a. 8 = 2 b. –1 = –1 c.–27 = –3
d. 16 = 2 e. –16 = not real f. 10 ≈ 2.15..
3 3 3
Higher Root
If r3 = x, then we say a is the cube root of x and we write this
as x = r. In general, if r k = x then we say r is the k’th root of x,
and we write it as a = x. In the cases of even roots,
i.e. k = 2, 4, 6, … we must have x > 0 so that x = a > 0.
4 4 3
67. Radicals and Pythagorean Theorem
3
k
k
Example E.
a. 8 = 2 b. –1 = –1 c.–27 = –3
d. 16 = 2 e. –16 = not real f. 10 ≈ 2.15..
3 3 3
Higher Root
If r3 = x, then we say a is the cube root of x and we write this
as x = r. In general, if r k = x then we say r is the k’th root of x,
and we write it as a = x. In the cases of even roots,
i.e. k = 2, 4, 6, … we must have x > 0 so that x = a > 0.
4 4 3
68. Here is one version of the proof of the Pythagorean Theorem.
Radicals and Pythagorean Theorem
69. Here is one version of the proof of the Pythagorean Theorem.
Radicals and Pythagorean Theorem
Given any right triangle, taking four copies of it and arrange
them as shown:
70. Here is one version of the proof of the Pythagorean Theorem.
Radicals and Pythagorean Theorem
Given any right triangle, taking four copies of it and arrange
them as shown:
71. Here is one version of the proof of the Pythagorean Theorem.
Radicals and Pythagorean Theorem
Given any right triangle, taking four copies of it and arrange
them as shown:
72. Here is one version of the proof of the Pythagorean Theorem.
Radicals and Pythagorean Theorem
Given any right triangle, taking four copies of it and arrange
them as shown:
73. Here is one version of the proof of the Pythagorean Theorem.
Radicals and Pythagorean Theorem
A
B
Given any right triangle, taking four copies of it and arrange
them as shown:
C
A + B
C
74. Here is one version of the proof of the Pythagorean Theorem.
Radicals and Pythagorean Theorem
A
B
The area of the large square is the sum of four triangles
and the smaller central squares.
Given any right triangle, taking four copies of it and arrange
them as shown:
C
A + B
C
75. Here is one version of the proof of the Pythagorean Theorem.
Radicals and Pythagorean Theorem
A
B
The area of the large square is the sum of four triangles
and the smaller central squares.
Given any right triangle, taking four copies of it and arrange
them as shown:
C
A + B
=
A
B
4 * +
C
C
C
A + B
76. Here is one version of the proof of the Pythagorean Theorem.
Radicals and Pythagorean Theorem
A
B
The area of the large square is the sum of four triangles
and the smaller central squares. Hence in area formulas that:
Given any right triangle, taking four copies of it and arrange
them as shown:
C
A + B
=
A
B
4 * +
C
C
C
A + B
(A+B)2 = 4* (
AB
2
) + C2
77. Here is one version of the proof of the Pythagorean Theorem.
Radicals and Pythagorean Theorem
A
B
The area of the large square is the sum of four triangles
and the smaller central squares. Hence in area formulas that:
Given any right triangle, taking four copies of it and arrange
them as shown:
C
A + B
=
A
B
4 * +
C
C
C
A + B
(A+B)2 = 4* (
AB
2
) + C2, so
A2 + 2AB + B2 = 2AB + C2
78. Here is one version of the proof of the Pythagorean Theorem.
Radicals and Pythagorean Theorem
A
B
The area of the large square is the sum of four triangles
and the smaller central squares. Hence in area formulas that:
Given any right triangle, taking four copies of it and arrange
them as shown:
C
A + B
=
A
B
4 * +
C
C
C
A + B
(A+B)2 = 4* (
AB
2
) + C2, so
A2 + 2AB + B2 = 2AB + C2
79. Here is one version of the proof of the Pythagorean Theorem.
Radicals and Pythagorean Theorem
A
B
The area of the large square is the sum of four triangles
and the smaller central squares. Hence in area formulas that:
Given any right triangle, taking four copies of it and arrange
them as shown:
C
A + B
=
A
B
4 * +
C
C
C
A + B
(A+B)2 = 4* (
AB
2
) + C2, so
A2 + 2AB + B2 = 2AB + C2
hence A2 + B2 = C2
80. Here is one version of the proof of the Pythagorean Theorem.
Radicals and Pythagorean Theorem
A
B
The area of the large square is the sum of four triangles
and the smaller central squares. Hence in area formulas that:
Given any right triangle, taking four copies of it and arrange
them as shown:
C
A + B
=
A
B
4 * +
C
C
C
A + B
(A+B)2 = 4* (
AB
2
) + C2, so
A2 + 2AB + B2 = 2AB + C2
hence A2 + B2 = C2
Using this formula for the right triangle with
A = B = 1
1
1
81. Here is one version of the proof of the Pythagorean Theorem.
Radicals and Pythagorean Theorem
A
B
The area of the large square is the sum of four triangles
and the smaller central squares. Hence in area formulas that:
Given any right triangle, taking four copies of it and arrange
them as shown:
C
A + B
=
A
B
4 * +
C
C
C
A + B
(A+B)2 = 4* (
AB
2
) + C2, so
A2 + 2AB + B2 = 2AB + C2
hence A2 + B2 = C2
Using this formula for the right triangle with
A = B = 1 so C2 = 12 + 12 = 2,
1
1
C = √2
82. Here is one version of the proof of the Pythagorean Theorem.
Radicals and Pythagorean Theorem
A
B
The area of the large square is the sum of four triangles
and the smaller central squares. Hence in area formulas that:
Given any right triangle, taking four copies of it and arrange
them as shown:
C
A + B
=
A
B
4 * +
C
C
C
A + B
(A+B)2 = 4* (
AB
2
) + C2, so
A2 + 2AB + B2 = 2AB + C2
hence A2 + B2 = C2
Using this formula for the right triangle with
A = B = 1 so C2 = 12 + 12 = 2, so that C = √2 ≈ 1.41...
1
1
C = √2
83. Here is one version of the proof of the Pythagorean Theorem.
Radicals and Pythagorean Theorem
A
B
The area of the large square is the sum of four triangles
and the smaller central squares. Hence in area formulas that:
Given any right triangle, taking four copies of it and arrange
them as shown:
C
A + B
=
A
B
4 * +
C
C
C
A + B
(A+B)2 = 4* (
AB
2
) + C2, so
A2 + 2AB + B2 = 2AB + C2
hence A2 + B2 = C2
Using this formula for the right triangle with
A = B = 1 so C2 = 12 + 12 = 2, so that C = √2 ≈ 1.41...
This number √2 is not a fraction so it's said to be irrational.
1
1
C = √2
85. Radicals and Pythagorean Theorem
(The strategy of this proof, to prove an impossibility, is to
exam all the possible forms of p/q if √2 = p/q as a fraction,
then argue that none of these forms is possible.
The Proof for "√2 is not a fraction"
86. Radicals and Pythagorean Theorem
(The strategy of this proof, to prove an impossibility, is to
exam all the possible forms of p/q if √2 = p/q as a fraction,
then argue that none of these forms is possible.
Hence √2 can’t be a fraction.)
The Proof for "√2 is not a fraction"
87. Radicals and Pythagorean Theorem
Proof: If √2 = p/q, as a fraction, is even/even, we may reduce it
to the following forms: odd/odd, odd/even or even/odd.
(The strategy of this proof, to prove an impossibility, is to
exam all the possible forms of p/q if √2 = p/q as a fraction,
then argue that none of these forms is possible.
Hence √2 can’t be a fraction.)
The Proof for "√2 is not a fraction"
88. Radicals and Pythagorean Theorem
Proof: If √2 = p/q, as a fraction, is even/even, we may reduce it
to the following forms: odd/odd, odd/even or even/odd.
We argue in two parts.
(The strategy of this proof, to prove an impossibility, is to
exam all the possible forms of p/q if √2 = p/q as a fraction,
then argue that none of these forms is possible.
Hence √2 can’t be a fraction.)
I. If √2 = or
odd
even
The Proof for "√2 is not a fraction"
odd
odd
89. Radicals and Pythagorean Theorem
Proof: If √2 = p/q, as a fraction, is even/even, we may reduce it
to the following forms: odd/odd, odd/even or even/odd.
We argue in two parts.
(The strategy of this proof, to prove an impossibility, is to
exam all the possible forms of p/q if √2 = p/q as a fraction,
then argue that none of these forms is possible.
Hence √2 can’t be a fraction.)
I. If √2 = or
odd
even
(√2) 2 =
squaring both sides, we’ve that
The Proof for "√2 is not a fraction"
odd
odd
orodd2
even2 odd2
odd2
90. Radicals and Pythagorean Theorem
Proof: If √2 = p/q, as a fraction, is even/even, we may reduce it
to the following forms: odd/odd, odd/even or even/odd.
We argue in two parts.
(The strategy of this proof, to prove an impossibility, is to
exam all the possible forms of p/q if √2 = p/q as a fraction,
then argue that none of these forms is possible.
Hence √2 can’t be a fraction.)
I. If √2 = or
odd
even
(√2) 2 =
squaring both sides, we’ve that
The Proof for "√2 is not a fraction"
odd
odd
orodd2
even2 odd2
odd2
2 = orodd
even odd
odd
Since odd2 is odd and even2 is even,
so
91. Radicals and Pythagorean Theorem
Proof: If √2 = p/q, as a fraction, is even/even, we may reduce it
to the following forms: odd/odd, odd/even or even/odd.
We argue in two parts.
(The strategy of this proof, to prove an impossibility, is to
exam all the possible forms of p/q if √2 = p/q as a fraction,
then argue that none of these forms is possible.
Hence √2 can’t be a fraction.)
I. If √2 = or
odd
even
(√2) 2 =
squaring both sides, we’ve that
The Proof for "√2 is not a fraction"
odd
odd
orodd2
even2 odd2
odd2
Clearing the denominator, we've that2 = orodd
even odd
odd
2*even = odd
Since odd2 is odd and even2 is even,
or 2*odd = odd
so
92. Radicals and Pythagorean Theorem
Proof: If √2 = p/q, as a fraction, is even/even, we may reduce it
to the following forms: odd/odd, odd/even or even/odd.
We argue in two parts.
(The strategy of this proof, to prove an impossibility, is to
exam all the possible forms of p/q if √2 = p/q as a fraction,
then argue that none of these forms is possible.
Hence √2 can’t be a fraction.)
I. If √2 = or
odd
even
(√2) 2 =
squaring both sides, we’ve that
The Proof for "√2 is not a fraction"
odd
odd
orodd2
even2 odd2
odd2
Clearing the denominator, we've that2 = orodd
even odd
odd
2*even = odd
Since odd2 is odd and even2 is even,
or 2*odd = odd
so
Both are impossible so √2 can’t be of the form or
odd
even odd .
odd
94. Radicals and Pythagorean Theorem
II. If √2 = p/q =
odd
even
is of the form 2*# (some number), so √2 = odd
2*#
.
Next we ague that p/q can’t be of the form
odd
even
and any even numbers
.
95. Radicals and Pythagorean Theorem
II. If √2 = p/q =
odd
even
Squaring both sides yields
2 =
odd2
is of the form 2*# (some number), so √2 =
(2*#)(2*#)
odd
2*#
.
.
Next we ague that p/q can’t be of the form
odd
even
and any even numbers
.
96. Radicals and Pythagorean Theorem
II. If √2 = p/q =
odd
even
Squaring both sides yields
2 =
odd2
is of the form 2*# (some number), so √2 =
(2*#)(2*#)
odd
2*#
Clearing the denominator, we've that
2*odd2 = (2*#)(2*#)
.
.
Next we ague that p/q can’t be of the form
odd
even
and any even numbers
.
97. Example C. Evaluate 2x3 – 5x2 + 2x
for x = -2, -1, 3 by factoring it first.
2x3 – 5x2 + 2x = x(2x2 – 5x + 2)
= x(2x – 1)(x – 2)
For x = -2:
(-2)[2(-2) – 1] [(-2) – 2] = -2 [-5] [-4] = -40
For x = -1:
(-1)[2(-1) – 1] [(-1) – 2] = -1 [-3] [-3] = -9
For x = 3:
3 [2(3) – 1] [(3) – 2] = 3 [5] [1] = 15
Applications of Factoring
Your turn: Double check these answers via the expanded form.
98. Radicals and Pythagorean Theorem
II. If √2 = p/q =
odd
even
Squaring both sides yields
2 =
odd2
is of the form 2*# (some number), so √2 =
(2*#)(2*#)
odd
2*#
Clearing the denominator, we've that
2*odd2 = (2*#)(2*#) This is impossible because
there is one 2 on the left but
only one 2
.
.
Next we ague that p/q can’t be of the form
odd
even
and any even numbers
.
99. Radicals and Pythagorean Theorem
II. If √2 = p/q =
odd
even
Squaring both sides yields
2 =
odd2
is of the form 2*# (some number), so √2 =
(2*#)(2*#)
odd
2*#
Clearing the denominator, we've that
2*odd2 = (2*#)(2*#) This is impossible because
there is one 2 on the left but
there’re two or more 2's to the right.only one 2 at least two 2’s
.
.
Next we ague that p/q can’t be of the form
odd
even
and any even numbers
.
100. Radicals and Pythagorean Theorem
II. If √2 = p/q =
odd
even
Squaring both sides yields
2 =
odd2
is of the form 2*# (some number), so √2 =
(2*#)(2*#)
odd
2*#
Clearing the denominator, we've that
2*odd2 = (2*#)(2*#) This is impossible because
there is one 2 on the left but
there’re two or more 2's to the right.only one 2 at least two 2’s
So √2 can’t be a fraction of the form odd .
even
.
.
Next we ague that p/q can’t be of the form
odd
even
and any even numbers
.
101. Radicals and Pythagorean Theorem
II. If √2 = p/q =
odd
even
Squaring both sides yields
2 =
odd2
is of the form 2*# (some number), so √2 =
(2*#)(2*#)
odd
2*#
Clearing the denominator, we've that
2*odd2 = (2*#)(2*#) This is impossible because
there is one 2 on the left but
there’re two or more 2's to the right.only one 2 at least two 2’s
So √2 can’t be a fraction of the form odd .
even
From part I and II, we see √2 fits none of the fractional forms.
Hence √2 is not a fraction.
.
.
Next we ague that p/q can’t be of the form
odd
even
and any even numbers
.
102. Radicals and Pythagorean Theorem
Exercise A. Solve for x. Give both the exact and approximate
answers. If the answer does not exist, state so.
1. x2 = 1 2. x2 – 5 = 4 3. x2 + 5 = 4
4. 2x2 = 31 5. 4x2 – 5 = 4 6. 5 = 3x2 + 1
7. 4x2 = 1 8. x2 – 32 = 42 9. x2 + 62 = 102
10. 2x2 + 7 = 11 11. 2x2 – 5 = 6 12. 4 = 3x2 + 5
x
3
4
Exercise B. Solve for x. Give both the exact and approximate
answers. If the answer does not exist, state so.
13. 4
3
x14. x
12
515.
x
1
116. 2
1
x17. 3 2
3
x18.
103. Radicals and Pythagorean Theorem
x
4
19.
x
x20.
3 /3
21.
43 5 2
6 /3
Exercise D. Find the exact answer.
3
–122. 23. 13
–12527.
8 24. –13
8
3
–2725.
26. –13
64
3
10028. 100029.
3
10,00030. 1,000,00031.
3
0.0132. 0.00133.
3
0.000134. 0.00000135.
3
x