1. Solving Quadratic Equations by the Quadratic Formula
8.2
Solving Quadratic Equations by the Quadratic Formula

2. The Quadratic Formula
Another technique for solving quadratic equations is to use the quadratic formula.
The formula is derived from completing the square of a general quadratic equation.

3. The Quadratic Formula Quadratic Formula
A quadratic equation written in the form ax2 + bx + c = 0 has the solutions

4. Example Solve 3x2 + x – 3 = 0 by the quadratic formula.
a = 3, b = 1, c = –3 4

5. Example
Solve 4x2 – 2x – 5 = 0 by the quadratic formula. 5

6. Example
Solve 11x2 – 9x = 1 by the quadratic formula. 6

7. Example
Solve x2 + x – = 0 by the quadratic formula. 7

8. Example
Solve x(x + 6) = –30 by the quadratic formula. 8

9. The Discriminant
The expression under the radical sign in the formula (b2 – 4ac) is called the discriminant.
b2 – 4ac
Number and Type of Solutions
Positive
Two real solutions
Zero
One real solution
Negative
Two complex but not real solutions

10. Example
Use the discriminant to determine the number and type of solutions for the following equation.
5 – 4x + 12x2 = 0

11. Example
Use the discriminant to determine the number and type of solutions for the following equation.
2x2 – 7x – 4 = 0

13. Example
At a local university, students often leave the sidewalk and cut across the lawn to save walking distance. Given the diagram below of a favorite place to cut across the lawn, approximate to the nearest foot how many feet of walking distance a student saves by cutting across the lawn instead of walking on the sidewalk.
Continued 13

14. Example (cont)
Use the Pythagorean theorem.
Continued