1. Algebra 1
Section 12.3

2. Solving Quadratic Equations
Some quadratic equations cannot be solved by factoring.
A method called “completing the square” can often be used when factoring cannot.

3. Completing the Square
To complete the square of an expression in the form x2 + bx, add the square of half the coefficient of x: x2 + bx + (b/2)2.

4. Example 1a x2 – 16x Since b = -16, (b/2)2 = (-8)2 = 64 x2 – 16x + 64

5. Example 1b x2 + x Since b = 3/5, (b/2)2 = (3/10)2 = 9/100 x2 + x +

6. Completing the Square
This method can be used to solve any quadratic equation of the form x2 + bx + c.
Remember to add the same number to both sides of the equation.

7. Example 2 Solve x2 + 8x = 9. Since b = 8, (b/2)2 = (4)2 = 16
Add 16 to both sides.
x2 + 8x + 16 =
x2 + 8x + 16 = 25
(x + 4)2 = 25

8. Be sure to check your solutions in the original equation.
Example 2
(x + 4)2 = 25
x + 4 = ±5
x = -4 ± 5
x = 1, -9
Be sure to check your solutions in the original equation.

9. Solving Quadratic Equations by Completing the Square
Write the equation in the form x2 + bx = c.
Complete the square by adding (b/2)2 to both sides of the equation.
Factor the trinomial.

10. Solving Quadratic Equations by Completing the Square
Apply the Square Root Property: If x2 = n, then x = ± n .
Solve the resulting equations.

11. Example 3 Solve a2 – 7a – 3 = 0. a2 – 7a = 3 (-7/2)2 = 49/4
Add 49/4 to both sides.
a2 – 7a + 49/4 = /4
(a – 7/2)2 = 61/4

12. In decimal form, the solution set is ≈ {-0.41, 7.41}.
Example 3
(a – 7/2)2 = 61/4
a – = ± 61 2 7
a =
7 ± 61 2
In decimal form, the solution set is ≈ {-0.41, 7.41}.

13. Example 4 Let x = the original length and width of the square garden
x + 4 = the new width
x + 7 = the new length
Using the area formula:
(x + 7)(x + 4) = 550

14. Example 4 (x + 7)(x + 4) = 550 x2 + 11x + 28 = 550 x2 + 11x = 522
121 4
(x )2 = 11 2
2209 4

15. Example 4 (x + )2 = x + = ± x + = ± x = 11 2 2209 4 11 2 2209 4 11 247
x =
-11 ± 47 2

16. Example 4
x =
-11 ± 47 2
x = 18 or -29
Since a negative answer is not reasonable, the original garden was 18 ft on each side.

17. Homework: pp