2 Solving EquaTIONS Using the Zero Product Rule Definition of a Quadratic Equation in One VariableIf a, b, and c are real numbers such that a ≠ 0, then a quadratic equation is an equation that can be written in the formax² + bx + c = 0
3 Zero Product Rule If ab = 0, then a = 0 or b = 0 Factor Apply the zero product ruleSet each factor equal to zeroSolve each equation for x
4 Steps for Solving a Quadratic Equation by Factoring Write the equation in the form:ax² + bx + c = 0Factor 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.
5 Solve Quadratic Equation Write the equation in the formax² + bx +c = 0Factor the polynomial completelySet each factor equal to zeroSolve each equationSolutions
6 Solve Quadratic Equation Write the equation in the formax² + bx +c = 0Factor the polynomial completelySet each factor equal to zeroSolve each equationSolutions
8 Solve Quadratic Equation Write the equation in the formax² + bx +c = 0Factor the polynomial completelySet each factor equal to zeroSolve each equationSolutions
9 Solve Higher Quadratic 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.Solve each equationSolutions
10 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) integerSimplifyFactor the polynomial completelySet each factor equal to zeroSolutionsSolve each equation
11 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 hypotenuseThe Pythagorean Theorem relates the lengths of the sides.chypotenusec2 = a2 + b2legbc2 = (3)2 + (4)2legac2 = = 25c = 5If one leg of a right triangle measures 3 inches and one leg measures 4 inches, how long is the hypotenuse?
12 Applying the Pythagorean Theorem Apply the Pythagorean theorem.c =10a² + b² = c²aSubstitute b= 6 and c = 10a² + 6² = 10²Simplifya² + 36 = 100a² =b =6a² - 64 = 0Factora = a = 8a + 8 =0 or a – 8 = 0(a + 8)(a - 8) = 0Set each factor equal to zeroBecause x represents the length of a side of a triangle, reject the negative solution.