1. 5.9 The Quadratic Formula
12/11/2013

2. Quadratic Formula:
Is used to solve quadratic equations that cannot be solved using the “Big X” method.
It can also be used instead of the “Big X”.
The equation must be written as:
ax2 + bx +c = 0.

3. Example 1 Solve the equation . - x 2 + 2x 15 = 0 = ( ) 3 x – 5 + = 3 x
Factor. = ( ) 3 x – 5 +
(mult)
-15 = 3 x – or 5 + 5 -3
+ 3 = + 3
- 5 = - 5 2
(add) = 3 x 5 – = x 2a b – +
4ac
b 2
a = b = 2 c = -15
𝑥= −2+8 2 = 6 2 =3 = x 2 – + 4 22 ( ) 1 15
𝑥= −2−8 2 = −10 2 =−5
𝑥= −2±
= −2±
= −2±8 2

4. Example 2 = x 2a b – + 4ac b 2 Solve . = x 2 + 3x 5 – SOLUTION = x 3 –
Solve an Equation with Two Real Solutions = x 2a b – +
4ac
b 2
Solve =
x 2 + 3x 5 –
SOLUTION = x 3 – + 4 32 ( ) 1 5 2
Substitute values in the quadratic formula: a 1, b 3, and c = x 3 – + 29 2
Simplify.
ANSWER
The solutions are and 3 – + 29 2 ≈
1.19
4.19

5. Use the quadratic formula to solve the equation.
Checkpoint
Solve an Equation with Two Real Solutions
Use the quadratic formula to solve the equation. 1. =
x 2 + 2x 3 – 3. =
3x 2 2x 6 –
ANSWER 3, – 1
ANSWER 1 19 3 ≈
1.79 +
1.12 – 2. =
2x 2 + 4x 1 –
ANSWER 2 – + 6 ≈
0.22,
2.22

6. Example 3 2a x = – b + b 2 4ac Solve x 2 2x 2 + 0. = SOLUTION x = + –
Solve an Equation with Imaginary Solutions 2a x = – b +
b 2
4ac
Solve
x 2 2x 2 + 0. =
SOLUTION x = + – 2 1 ( 4 22
Substitute values in the quadratic formula: a 1,
b , and c x = + – 2 4
Simplify. – 2 + – 2i
Simplify and rewrite
using the imaginary unit i. x = 2
Simplify. x = + – 1 i + – 1 i
ANSWER
The solutions are and 6

7. Use the quadratic formula to solve the equation.
Checkpoint
Use the Quadratic Formula
Use the quadratic formula to solve the equation.
ANSWERS 4.
x 2 4x 5 + = + – 2 i
x 2 2x 3 + = – 5. + – 1 i 2 6.
2x 2 x = 4 – + + – 1 i 4 31

8. Homework:
5.9 p.278 #31-40