1. Objectives: 1. Solve equations by: A. Factoring B. Square Root of Both Sides C. Completing the Square D. Quadratic Formula 2. Solve equations in quadratic form. Objectives: 1. Solve equations by: A. Factoring B. Square Root of Both Sides C. Completing the Square D. Quadratic Formula 2. Solve equations in quadratic form. 2.2

2. Definition of a Quadratic Equation An equation that can be written in the form ax 2 + bx + c = 0 with real constants a, b, & c, with a ≠ 0. Techniques to solve a quadratic equation:  Techniques that sometimes work Factoring Taking the square root of both sides of an equation  Techniques that always work Completing the Square Using the Quadratic Formula

3. Example #1a Solving by Factoring Solve by Factoring:

4. Example #1b Solving by Factoring Solve by Bottoms Up Method: Don’t forget to first factor out the GCF, if necessary. To factor this trinomial you must first multiply a by c, factor normally, divide both factors by a & reduce, and finally bring “bottoms up”.

5. Example #2 Solving ax 2 = b Solve by Taking the Square Root of Both Sides:

6. Example #3 Solvinga(x − h) 2 = k Solve by Taking the Square Root of Both Sides:

7. Example #4 Solving a Quadratic Equation by Completing the Square Solve by Completing the Square: Completing the square only works when the coefficient of x 2 is a 1. Always divide every term by the leading coefficient before attempting to complete the square.

8. Example #4 Solving a Quadratic Equation by Completing the Square Solve by Completing the Square: Remember after completing the square, add it to both sides of the equation.

9. The Quadratic Formula

10. Example #5 Solving a Quadratic Equation by Using the Quadratic Formula Solve by Using the Quadratic Formula: Quadratic equations must be in Standard Form ax 2 + bx + c = 0 before using the Quadratic Formula.

11. The Discriminant Discriminant is PositiveTwo Real Solutions Discriminant is ZeroOne Real Solution Discriminant is NegativeNo Real Solution The discriminant is used to determine the number of solutions without solving the problem.

12. Example #6 Determining the Number of Solutions by Using the Discriminant Find the Number of Solutions: − 31 is NOT a solution to the equation. Since the discriminant is negative, this means that there is no real solution to the equation.

13. Example #7 Polynomials in Quadratic Form Solving an Equation in Quadratic Form: Compare this polynomial equation with the quadratic form: ax 2 + bx + c = 0 This polynomial can be factored using Bottoms Up!