1. Quadratic Equations By Dr. Carol A. Marinas

2. Solving Equations In the previous section, we solved LINEAR equations. This means that the highest exponent is 1. In this section, we will be solving equations with the highest exponent of 2. These are called Quadratic Equations.

3. What are Quadratic Equations? They are in the form: ax 2 + bx + c = 0 Methods of Solving Quadratic Equations (Put into above form first)? –By Factoring (sometimes works) –By Quadratic Formula (always works)

4. By Factoring Put in Standard Form Factor the polynomial Set each factor to 0 Solve for the variable Check answer in original equation x 2 - 3x = 4 x 2 - 3x - 4 = 0 (x - 4) (x + 1) = 0 x - 4 = 0 or x + 1 = 0 x = 4 or x = -1 4 2 - 3(4) = 4 (-1) 2 - 3(-1) = 4

5. By Quadratic Formula This method works all the time. It is usually used when the Factoring method fails. Put in standard form. Determine the values of a, b, and c.

6. The Quadratic Formula _______ -b +  b 2 - 4ac x = ------------------------ 2 a

7. Example: x 2 + 9x = -8 Standard form: x 2 + 9x + 8 = 0 We could do this by Factoring but let’s use the Quadratic Formula. a = 1, b = 9 and c = 8 -9 +  9 2 - 4(1)(8) -9 +  49 x = ------------------------ = ----------- 2(1) 2

8. Example Continues -9 +  49 -9 + 7 x = -------------- = ------- 2 2 x = (-9 - 7)/2 or x = (-9 + 7)/2 x = -8 or x = -1 x 2 + 9x = -8 Check: (-8) 2 + 9(-8) = -8 (-1) 2 + 9(-1) = -8

9. REMEMBER --- If you have a “squared” term, it is a “quadratic” equation. You must put the “quadratic equation” into standard form first. This means that all terms are on the left side and the right side is 0. Not necessary but I would multiply through to eliminate the fractions. Use either the “factoring” or “quadratic formula” method to solve. Finally, check your answers in the original equation.