Forms of a Quadratic Equation x(x - 2) = 0 -4x² + 4x =1 -4x² + 4x -1 = 0 x² - 2x = 0 (x – 4)(x + 4) = 9 x² - 16 = 9 x² - 25 = 0

Solving EquaTIONS Using the Zero Product Rule Definition of a Quadratic Equation in One Variable If a, b, and c are real numbers such that a ≠ 0, then a quadratic equation is an equation that can be written in the form ax² + bx + c = 0

Zero Product Rule If ab = 0, then a = 0 or b = 0 Factor Apply the zero product rule Set each factor equal to zero Solve each equation for x

Steps for Solving a Quadratic Equation by Factoring Write the equation in the form: ax² + bx + c = 0 Factor the equation completely. Apply the zero product rule, that is, set each factor equal to zero, and solve the resulting equations. Note: The solution(s) found in Step 3 may be checked by substitution into the original equation.

Solve Quadratic Equation Write the equation in the form ax² + bx +c = 0 Factor the polynomial completely Set each factor equal to zero Solve each equation Solutions

Check: x = -½ Check: x = 5

Solve Quadratic Equation Write the equation in the form ax² + bx +c = 0 Factor the polynomial completely Set each factor equal to zero Solve each equation Solutions

Solve Higher Degree Polynomial Equation This is a higher degree polynomial equation. The equation is already set equal to zero. Because there are four terms, try factoring by grouping. Set each factor equal to zero Solve each equation Solutions

Translating to Quadratic Equation The product of two consecutive integers is 48 more than the larger integer. Find the integers. Let x represent the first (smaller) integer. Then x + 1 represents the second (larger) integer Simplify Factor the polynomial completely Set each factor equal to zero Solve each equation Solutions

c2 = a2 + b2 c c2 = (3)2 + (4)2 a c2 = 9 + 16 = 25 c = 5 In a right triangle, the shorter sides are called legs and the longest side (which is the one opposite the right angle) is called the hypotenuse The Pythagorean Theorem relates the lengths of the sides. c hypotenuse c2 = a2 + b2 leg b c2 = (3)2 + (4)2 leg a c2 = 9 + 16 = 25 c = 5 If one leg of a right triangle measures 3 inches and one leg measures 4 inches, how long is the hypotenuse?

Applying the Pythagorean Theorem Apply the Pythagorean theorem. c =10 a² + b² = c² a Substitute b= 6 and c = 10 a² + 6² = 10² Simplify a² + 36 = 100 a² + 36 -100 = 100 - 100 b =6 a² - 64 = 0 Factor a = -8 a = 8 a + 8 =0 or a – 8 = 0 (a + 8)(a - 8) = 0 Set each factor equal to zero Because x represents the length of a side of a triangle, reject the negative solution.