1. Section 8.4
Quadratic Formula

2. Objectives Solving Quadratic Equations The Discriminant
Quadratic Equations Having Complex Solutions

3. QUADRATIC FORMULA
The solutions to ax2 + bx + c = 0 with a ≠ 0 are given by

4. Example
Solve the equation 4x2 + 3x – 8 = 0. Support your results graphically. Solution Symbolic Solution Let a = 4, b = 3 and c = − 8. or or

5. Example (cont)
4x2 + 3x – 8 = 0 Graphical Solution

6. Example
Solve the equation 3x2 − 6x + 3 = 0. Support your result graphically. Solution Let a = 3, b = −6 and c = 3.

7. Example
Solve the equation 2x2 + 4x + 5 = 0. Support your result graphically. Solution Let a = 2, b = 4 and c = 5.
There are no real solutions
for this equation because
is not a real number.

8. THE DISCRIMINANT AND QUADRATIC
EQUATIONS
To determine the number of solutions to the quadratic equation ax2 + bx + c = 0, evaluate the discriminant b2 – 4ac.
1. If b2 – 4ac > 0, there are two real solutions.
2. If b2 – 4ac = 0, there is one real solution.
3. If b2 – 4ac < 0, there are no real solutions; there are two complex solutions.

9. Example
Use the discriminant to determine the number of solutions to −2x2 + 5x = 3. Then solve the equation using the quadratic formula. Solution −2x2 + 5x − 3 = 0 Let a = −2, b = 5 and c = −3.
b2 – 4ac
= (5)2 – 4(−2)(−3)
= 1 or
Thus, there are two solutions.

10. THE EQUATION x2 + k = 0
If k > 0, the solution to x2 + k = 0 are given by

11. Example
Solve x = 0. Solution The solutions are

12. Example
Solve 3x2 – 7x + 5 = 0. Write your answer in standard form: a + bi. Solution Let a = 3, b = −7 and c = 5.
and

13. Example
Solve Write your answer in standard form: a + bi. Solution Begin by adding 2x to each side of the equation and then multiply by 5 to clear fractions. Let a = −2, b = 10 and c = −15.

14. Example (cont)
Let a = −2, b = 10 and c = −15.

15. Example
Solve by completing the square. Solution After applying the distributive property, the equation becomes Since b = −4 ,add to each side of the equation.
The solutions are 2 + i and 2 − i.