* Use the Pythagorean Theorem to solve problems * Factor quadratic expressions and solve quadratic equations by factoring

1. The Pythagorean Theorem
The student is able to (I can):
• Use the Pythagorean Theorem to solve problems.
• Factor quadratic expressions and solve quadratic
equations by factoring.

2. The Pythagorean Theorem (a2 + b2 = c2) states the
relationship between the sides of a right triangle. Although it
was named for Pythagoras (circa 500 B.C.), this relationship
was actually known to earlier people, including the
Babylonians, Egyptians, and the Chinese.
A Babylonian tablet from 1800 B.C. that is
presumed to be listing sides of right triangles.

3. The Pythagorean Theorem states that the sum of the squares
of the sides of a right triangle (a and b) is equal to the square
of the hypotenuse (c) or
The Pythagorean Theorem allows us to find an unknown side
of a right triangle if we know the other two sides.
Remember: the hypotenuse (opposite the right angle) is
always c.
2 2 2
a b c 
hypotenuse
right
angle

4. Example: Find the value of x.
x
12
13

5. Example: Find the value of x.
x
12
13
 2 2 2
12 13x

6. Example: Find the value of x.
x
12
13
 2 2 2
12 13x
 

 
2
2
144 169
25
25 5
x
x
x

7. Square Roots
• When we are taking the square root of a number, we will
not always get a whole number answer.
• If your answer is not a whole number, then the number
your calculator gives you is a decimal approximation. This
is an irrational number, like , which goes on forever and
does not repeat.
• If I ask for an exact answer, I do not want a decimal – I
want you to leave it as a radical.

8. Examples
Find the value of x. Leave any non-integer answers as
radicals.
1.
2.
2
6
x
x x-2
4

9. Examples
Find the value of x. Leave any non-integer answers as
radicals.
1.
2.
2 2 2
2 6 x 
2
4 36 x 
2
6
x
x x-2
4

10. Examples
Find the value of x. Leave any non-integer answers as
radicals.
1.
2.
2 2 2
2 6 x 
2
4 36 x 
2
40 x
40x 
2
6
x
x x-2
4

11. Examples
Find the value of x. Leave any non-integer answers as
radicals.
1.
2.
2 2 2
2 6 x 
2
4 36 x 
2
40 x
40x 
2 2 2
4 ( 2)x x  
2
6
x
x x-2
4

12. Examples
Find the value of x. Leave any non-integer answers as
radicals.
1.
2.
2 2 2
2 6 x 
2
4 36 x 
2
40 x
40x 
2 2 2
4 ( 2)x x   x -2
x x2 -2x
-2 -2x 4
2 2
16 4 4x x x   
2
6
x
x x-2
4

13. Examples
Find the value of x. Leave any non-integer answers as
radicals.
1.
2.
2 2 2
2 6 x 
2
4 36 x 
2
40 x
40x 
2 2 2
4 ( 2)x x   x -2
x x2 -2x
-2 -2x 4
2 2
16 4 4x x x   
20 – 4x = 0
20 = 4x
x = 5
2
6
x
x x-2
4

14. A quadratic expression is an expression in which the largest
exponent is 2. A quadratic equation is an equation which can
be written as ax2 + bx + c = 0.
As we saw from the last example, multiplying two binomials
together will produce a quadratic expression. When we
factor a quadratic expression, we go the other direction.
Generally, there are a couple of different ways to do this.
Method #1 (a = 1)
• Draw a large X. On top of the X, write the value of ac. On
the bottom, write the value of b. For the two sides, find
two number that multiply to c and add to b.
• Your two sets of parentheses will be x plus the two values.

15. Example: Factor x2 ‒ 7x + 12.

16. Example: Factor x2 ‒ 7x + 12.
• a = 1, b = ‒7, c = 12

17. Example: Factor x2 ‒ 7x + 12.
• a = 1, b = ‒7, c = 12
• Draw an X

18. Example: Factor x2 ‒ 7x + 12.
• a = 1, b = ‒7, c = 12
• Draw an X
• Put the 12 on top and the ‒7 on the bottom.
12
‒7

19. Example: Factor x2 ‒ 7x + 12.
• a = 1, b = ‒7, c = 12
• Draw an X
• Put the 12 on top and the ‒7 on the bottom.
• Two numbers that multiply to 12 and add to ‒7 are
‒3 and ‒4. They go on the two sides (it doesn’t
matter which comes first).
12
‒7
‒3‒4

20. Example: Factor x2 ‒ 7x + 12.
• a = 1, b = ‒7, c = 12
• Draw an X
• Put the 12 on top and the ‒7 on the bottom.
• Two numbers that multiply to 12 and add to ‒7 are
‒3 and ‒4. They go on the two sides (it doesn’t
matter which comes first).
• Create a pair of parentheses with x in each one along
with the two numbers. This is your answer.
12
‒7
‒3‒4
  4 3x x 

21. Method #1 (a > 1)
• If a is greater than 1, after you have found the two
numbers that multiply to c and add to b, create a box
similar to the one we used to multiply the two binomials
in the triangle example.
• This time, you will be filling in the inside of the box. Put
the first term in the first interior box and the last term in
the last interior box.
• For the upper right and lower left boxes, enter the two
values you found in step 1. Then, factor out the greatest
common factor of each row and column.
• The upper row and outer column will give you the two
factors of the expression.

22. Example: Factor 2x2 + x ‒ 6.

23. Example: Factor 2x2 + x ‒ 6.
• a = 2, b = 1, c = ‒6
• Draw an X and fill it in as before.
Remember to multiply a and c.
‒12
1
‒34

24. Example: Factor 2x2 + x ‒ 6.
• a = 2, b = 1, c = ‒6
• Draw an X and fill it in as before.
Remember to multiply a and c.
• Draw a box and fill in the interior spaces with your
values.
‒12
1
‒34
2x2 4x
‒3x ‒6

25. Example: Factor 2x2 + x ‒ 6.
• a = 2, b = 1, c = ‒6
• Draw an X and fill it in as before.
Remember to multiply a and c.
• Draw a box and fill in the interior spaces with your
values.
• Factor out the GCF of each row
and column.
‒12
1
‒34
x 2
2x 2x2 4x
-3 ‒3x ‒6

26. Example: Factor 2x2 + x ‒ 6.
• a = 2, b = 1, c = ‒6
• Draw an X and fill it in as before.
Remember to multiply a and c.
• Draw a box and fill in the interior spaces with your
values.
• Factor out the GCF of each row
and column.
• Write out the two factors.
‒12
1
‒34
x 2
2x 2x2 4x
-3 ‒3x ‒6
  2 3 2x x 

27. Method #2 (a > 1)
• Another method you can use works along the same lines.
Instead of drawing a box, you write out the expression and
break the middle term into the two values you found.
• Group the first two terms and the last two terms.
• Find the GCF of each group. The remaining part should be
the same for each group.
• Factor out this remaining part. This should give you your
two factors.

28. Example: Factor 6x2 ‒ 5x ‒ 4.

29. Example: Factor 6x2 ‒ 5x ‒ 4.
• a = 6, b = ‒5, c = ‒4.
• Draw an X and fill it in as before.
Remember to multiply a and c.
‒24
‒5
3‒8

30. Example: Factor 6x2 ‒ 5x ‒ 4.
• a = 6, b = ‒5, c = ‒4.
• Draw an X and fill it in as before.
Remember to multiply a and c.
• Write out the expression, breaking up the b term:
‒24
‒5
3‒8
2
6 8 3 4x x x  

31. Example: Factor 6x2 ‒ 5x ‒ 4.
• a = 6, b = ‒5, c = ‒4.
• Draw an X and fill it in as before.
Remember to multiply a and c.
• Write out the expression, breaking up the b term:
• Group the terms:
‒24
‒5
3‒8
2
6 8 3 4x x x  
   2
6 8 3 4x x x  

32. Example: Factor 6x2 ‒ 5x ‒ 4.
• a = 6, b = ‒5, c = ‒4.
• Draw an X and fill it in as before.
Remember to multiply a and c.
• Write out the expression, breaking up the b term:
• Group the terms:
• Factor out the GCF:
‒24
‒5
3‒8
2
6 8 3 4x x x  
   2
6 8 3 4x x x  
   2 3 4 1 3 4x x x  

33. Example: Factor 6x2 ‒ 5x ‒ 4.
• a = 6, b = ‒5, c = ‒4.
• Draw an X and fill it in as before.
Remember to multiply a and c.
• Write out the expression, breaking up the b term:
• Group the terms:
• Factor out the GCF:
• The 3x ‒ 4 is a common factor for both, so we can
factor that for our final answer:
‒24
‒5
3‒8
2
6 8 3 4x x x  
   2
6 8 3 4x x x  
   2 3 4 1 3 4x x x  
  3 4 2 1x x 

34. Method #3
• Guess and check – which pair of factors will produce the
given expression.
Solving Quadratic Equations by Factoring
• If a quadratic equation can be factored, this is the fastest
method.
• The most important step is to make sure the equation is
written in the form ax2 + bx + c = 0.
• Factor using the method of your choice.
• The only way the product of two numbers equals 0 is if
one or both factors equals 0. Therefore, split the factors
apart and set each factor equal to 0.
• Solve each equation.

35. Example: Solve 6x2 ‒ 5x ‒ 4 = 0.
• We factored this earlier, so we can write the equation as:
• Set each factor equal to 0 and solve:
• You can also write your answer using set notation:
  3 4 2 1 0x x  
3 4 0x   2 1 0x  
3 4
4
3
x
x


2 1
1
2
x
x
 
 
4 1
,
3 2
 
 
 