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. 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. 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. Example
Solve by factoring:
2
3 10x x− =

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. 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. 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. Try This
Solve by factoring:
2
18 23 6x x= +
2 3
or
9 2
x x= − =
Hint: write in standard form

9. Example
Solve by taking the square
root of both sides:
2
4 16x =a.
2
2 15x =b.

10. Try This
Solve by taking the square
root of both sides:
2
4 16x =
2x = ±

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. Example
Solve by taking the square
root of both sides:
What is wrong with this?
2
4x = −

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. Important Idea
Completing the square is the
process of finding the
number that will make the
expression a perfect square
trinomial.

15. Important Idea
2
12 36x x+ +
is a perfect square trinomial
because it factors as:
2
( 6)( 6) ( 6)x x x+ + = +

16. Try This
Complete the square for:
2
8x x+
then factor your result
2 2
8 16 ( 4)x x x+ + = +

17. Example
Complete the square for:
2 3
4
y y+
then factor your result.
Use fractions only.

18. Example
Solve by completing the
square:
2
8 14 0x x+ + =
1. Move the constant to the
right:
2
8 14x x+ = −

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. 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. Try This
Solve by completing the
square:
2
4 1 0x x− + =
2 3x = ±

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. Try This
Solve by completing the
square…fractions only.
2
2 13 15 0x x+ + =
3
, 5
2
x = − −

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. 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. 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. 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. Example
Solve using the quadratic
formula. Leave answer in
simplified radical form.
2
2 1x x= +

29. Try This
Solve using the quadratic
formula. Leave answer in
simplified radical form.
2
4 3 5x x− =
3 89
8
x
±
=

30. Example
Solve using the quadratic
formula. Leave answer in
simplified radical form.
4 2
4 13 3 0x x− + =

31. Lesson Close
State the quadratic
formula from memory.