2-2: Solving Quadratic
Equations Algebraically
© 2007 Roy L. Gover (www.mrgover.com)
Learning Goals:
•Solve by factoring
•...
Definition
A quadratic, or second
degree equation is one that
can be written in the form
2
0ax bx c+ + =
for real constant...
Important Idea
There are 4 techniques to
algebraically solve quadratic
equations:
•Factoring
•Taking square root of both
s...
Example
Solve by factoring:
2
3 10x x− =
Definition
The zero product property:
If the product of real
numbers is zero, then one
or both of the numbers
must be zero
Example
Solve by factoring:
What is wrong with this:
2
6
( 1) 6
6 & 1 6 7
x x
x x
x x x
− =
− =
= − = ⇒ =
Try This
Solve by factoring:
2
2 3 1 0t t+ + =
1
& 1
2
t t= − = −
Can you think of a way to
check your answer?
Try This
Solve by factoring:
2
18 23 6x x= +
2 3
or
9 2
x x= − =
Hint: write in standard form
Example
Solve by taking the square
root of both sides:
2
4 16x =a.
2
2 15x =b.
Try This
Solve by taking the square
root of both sides:
2
4 16x =
2x = ±
Try This
Solve by taking the square
root of both sides. Give exact
and approximate solutions.
2
3 16x =
4 3
2.309
3
x = ± ...
Example
Solve by taking the square
root of both sides:
What is wrong with this?
2
4x = −
Example
Complete the square for:
2
12x x+
1. Half the coefficient of x:
1/2 of 12=6
2. Square this number and
add to the e...
Important Idea
Completing the square is the
process of finding the
number that will make the
expression a perfect square
t...
Important Idea
2
12 36x x+ +
is a perfect square trinomial
because it factors as:
2
( 6)( 6) ( 6)x x x+ + = +
Try This
Complete the square for:
2
8x x+
then factor your result
2 2
8 16 ( 4)x x x+ + = +
Example
Complete the square for:
2 3
4
y y+
then factor your result.
Use fractions only.
Example
Solve by completing the
square:
2
8 14 0x x+ + =
1. Move the constant to the
right:
2
8 14x x+ = −
Example
Solve by completing the
square:
2
8 14 0x x+ + =
2. Complete the square and
add to the left and right:
2
8 16 14 1...
Example
2
( 4) 2x + =
4 2x + = ±
Solve by completing the
square: 2
8 14 0x x+ + =
3. Factor left side and solve:
4 2x = − ±
Try This
Solve by completing the
square:
2
4 1 0x x− + =
2 3x = ±
Example
Solve by completing the
square…
2
6 2 0x x− − =
Before you complete the
square, the coefficient of
the squared ter...
Try This
Solve by completing the
square…fractions only.
2
2 13 15 0x x+ + =
3
, 5
2
x = − −
Definition
The solutions to
2
0ax bx c+ + =
are:
2
4
2
b b ac
x
a
− ± −
=
These solutions are called
the Quadratic Formula
Important Idea
2
4
2
b b ac
x
a
− ± −
=
2
4b ac− is called the
discriminant
1. If 2
4 0b ac− >
there are 2 real solutions
Important Idea
2
4
2
b b ac
x
a
− ± −
=
2
4b ac− is called the
discriminant
2. If 2
4 0b ac− =
there is 1 real solution
Important Idea
2
4
2
b b ac
x
a
− ± −
=
2
4b ac− is called the
discriminant
3. If 2
4 0b ac− <
there are no real solutions
Example
Solve using the quadratic
formula. Leave answer in
simplified radical form.
2
2 1x x= +
Try This
Solve using the quadratic
formula. Leave answer in
simplified radical form.
2
4 3 5x x− =
3 89
8
x
±
=
Example
Solve using the quadratic
formula. Leave answer in
simplified radical form.
4 2
4 13 3 0x x− + =
Lesson Close
State the quadratic
formula from memory.
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Hprec2 2

  1. 1. 2-2: Solving Quadratic Equations Algebraically © 2007 Roy L. Gover (www.mrgover.com) Learning Goals: •Solve by factoring •Solve by taking square root of both sides •Solve by completing the square •Solve by using quadratic formula
  2. 2. Definition A quadratic, or second degree equation is one that can be written in the form 2 0ax bx c+ + = for real constants a,b, and c with a≠0. This is the standard form for a quadratic equation.
  3. 3. Important Idea There are 4 techniques to algebraically solve quadratic equations: •Factoring •Taking square root of both sides •Completing the square •Using quadratic formula
  4. 4. Example Solve by factoring: 2 3 10x x− =
  5. 5. Definition The zero product property: If the product of real numbers is zero, then one or both of the numbers must be zero
  6. 6. Example Solve by factoring: What is wrong with this: 2 6 ( 1) 6 6 & 1 6 7 x x x x x x x − = − = = − = ⇒ =
  7. 7. Try This Solve by factoring: 2 2 3 1 0t t+ + = 1 & 1 2 t t= − = − Can you think of a way to check your answer?
  8. 8. Try This Solve by factoring: 2 18 23 6x x= + 2 3 or 9 2 x x= − = Hint: write in standard form
  9. 9. Example Solve by taking the square root of both sides: 2 4 16x =a. 2 2 15x =b.
  10. 10. Try This Solve by taking the square root of both sides: 2 4 16x = 2x = ±
  11. 11. Try This Solve by taking the square root of both sides. Give exact and approximate solutions. 2 3 16x = 4 3 2.309 3 x = ± = ±
  12. 12. Example Solve by taking the square root of both sides: What is wrong with this? 2 4x = −
  13. 13. Example Complete the square for: 2 12x x+ 1. Half the coefficient of x: 1/2 of 12=6 2. Square this number and add to the expression 36+
  14. 14. Important Idea Completing the square is the process of finding the number that will make the expression a perfect square trinomial.
  15. 15. Important Idea 2 12 36x x+ + is a perfect square trinomial because it factors as: 2 ( 6)( 6) ( 6)x x x+ + = +
  16. 16. Try This Complete the square for: 2 8x x+ then factor your result 2 2 8 16 ( 4)x x x+ + = +
  17. 17. Example Complete the square for: 2 3 4 y y+ then factor your result. Use fractions only.
  18. 18. Example Solve by completing the square: 2 8 14 0x x+ + = 1. Move the constant to the right: 2 8 14x x+ = −
  19. 19. Example Solve by completing the square: 2 8 14 0x x+ + = 2. Complete the square and add to the left and right: 2 8 16 14 16x x+ + = − +
  20. 20. Example 2 ( 4) 2x + = 4 2x + = ± Solve by completing the square: 2 8 14 0x x+ + = 3. Factor left side and solve: 4 2x = − ±
  21. 21. Try This Solve by completing the square: 2 4 1 0x x− + = 2 3x = ±
  22. 22. Example Solve by completing the square… 2 6 2 0x x− − = Before you complete the square, the coefficient of the squared term must be 1
  23. 23. Try This Solve by completing the square…fractions only. 2 2 13 15 0x x+ + = 3 , 5 2 x = − −
  24. 24. Definition The solutions to 2 0ax bx c+ + = are: 2 4 2 b b ac x a − ± − = These solutions are called the Quadratic Formula
  25. 25. Important Idea 2 4 2 b b ac x a − ± − = 2 4b ac− is called the discriminant 1. If 2 4 0b ac− > there are 2 real solutions
  26. 26. Important Idea 2 4 2 b b ac x a − ± − = 2 4b ac− is called the discriminant 2. If 2 4 0b ac− = there is 1 real solution
  27. 27. Important Idea 2 4 2 b b ac x a − ± − = 2 4b ac− is called the discriminant 3. If 2 4 0b ac− < there are no real solutions
  28. 28. Example Solve using the quadratic formula. Leave answer in simplified radical form. 2 2 1x x= +
  29. 29. Try This Solve using the quadratic formula. Leave answer in simplified radical form. 2 4 3 5x x− = 3 89 8 x ± =
  30. 30. Example Solve using the quadratic formula. Leave answer in simplified radical form. 4 2 4 13 3 0x x− + =
  31. 31. Lesson Close State the quadratic formula from memory.

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