Section 11 1 Quadratic Equations Quadratic Equations The

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 – ● 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=0 OR ax + b = 0 Solve each equation – ● x(ax + b) = 0 Use the zero product property – ● ax² + bx = 0 x=0 OR Check your answers x = -b/a

Example ● ● Solve 2 x² – 3 = 29 – 2 x² = 29 +3 – 2 x² = 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 – For any real number k, if x² = k, the – x = √k – x = ± √k or x = -√k Theory – Let us factor this problem. – x² – 4 = 0 – x– 2=0 – x=2 (x – 2)(x + 2) = 0 OR OR x+2=0 x = -2

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

Example ● Solve 2 x² = 18

Example ● Solve -x² = 25

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

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

Generic Third Term ● Can you find a pattern to create 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 ● 2 x² + 8 x + 3 = 0 ● 2 x² + 8 x = -3 ● x² + 4 x = -3/4 – New C = [(1/2)(4)]² = 4 ● x² + 4 x + 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 + 6 x = 25

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

Homework ● 11. 1 – # 10, 11, 24, 39 -50, 51, 53, 55, 57