Section 11.1 Quadratic Equations

Quadratic Equations The general form of the quadratic equation is ax² + bx + c = 0 where a ≠ 0 The unique quadratic equation forms are ax² + bx + c = 0 ax² + bx = 0 ax² + c = 0 OR ax² - c = 0

Zero Product Property Zero Product Rule Theory If ab = 0 the a = 0 or b = 0 Theory How can you create a zero through multiplication? 0 * 0 # * 0 0 * #

How to solve ax² + bx = 0 Given Factor the Greatest Common Factor x(ax + b) = 0 Use the zero product property x = 0 OR ax + b = 0 Solve each equation x = 0 OR x = -b/a Check your answers

Example Solve 2x² – 3 = 29 Check 2x² = 29 +3 2x² = 32 x² = 16 x = 4 OR x = -4 Check x = 4 2(4)² – 3 = 29 29=29 x = -4 2(-4)² – 3 = 29 29=29

Example Solve x² – 1 = 15

Example Solve x² + 5 = 10

Square Root Property Square Root Property Theory For any real number k, if x² = k, the x = √k or x = -√k x = ± √k Theory Let us factor this problem. x² – 4 = 0 (x – 2)(x + 2) = 0 x – 2 = 0 OR x + 2 = 0 x = 2 OR x = -2

How to solve ax² - c = 0 Given Move the constant to the other side Divide both sides by the coefficient x² = c/a Use the square root property X = √c/a or -√c/a Check the solutions

Example Solve -5x² = -50 Check -5(x)² = -50 (x)² = 10 x = √10 OR x = -√10 Check x = √10 -5(√10)² = -5(10) = -50 x = -√10 -5(-√10)² = -5(10) = -50

Example Solve 2x² = 18

Example Solve -x² = 25

Example Solve 2x² = 18

How to solve ax² + bx + c = 0 There are three ways to solve this problem Factoring (5.7) Completing the square (11.2) Quadratic Formula (11.2)

Completing the square Completing the square takes the quadratic equation and turns it into a perfect square trinomial. Then you isolate the variable by using the square root property.

Perfect Square Trinomial x² + 2xb + b² = (x + b)² x² - 2xb + b² = (x - b)² We will be creating a perfect square trinomial We will start with the x² - 2xb We will create the third term We will then factor the perfect square trinomial

Examples of making the third term Find the constant for x² + 2x Find the constant for x² + 4x Find the constant for x² + 12x Find the constant for x² – 8x Find the constant for x² + 21x

Generic Third Term Can you find a pattern to create the third term or new constant?

Steps Write equation in decreasing order Move constant to the other side Divide both sides by the leading coefficient Find the new constant C = [(1/2)b]² Add the new constant to both sides Factor one side, simplify the other Use the square root property Isolate the variable

Solve by Completing the Square 2x² + 8x + 3 = 0 2x² + 8x = -3 x² + 4x = -3/4 New C = [(1/2)(4)]² = 4 x² + 4x + 4 = -3/4 + 4 (x+2)² = 13/4 X+2 = ±√(13/4) X+2 = ±√13 / 2 X = -2 ±√13 / 2

Solve by Completing the Square X² + 6x = 25

Solve by Completing the Square 3X² + 6x = 24

Homework 11.1 # 10, 11, 21, 24, 39-50, 51, 53, 55, 57