(touches x-axis twice) (does not touch x-axis) Section 10-5 Factor to Solve Quadratic Equations SPI 23E: Find the solution to a quadratic equation given in standard form SPI 23F: select the solution to a quadratic equation given solutions represented in graphical form Objectives: Solve (find real number solutions) by factoring The solutions of a quadratic equation are: the x-intercepts since they are on the real number line a quadratic equation can have solutions as follows: Two Solutions (touches x-axis twice) One Solutions (touches x-axis once) No Solution (does not touch x-axis)

Methods of Finding Real Number Solutions Two Methods for Solving Quadratic Equations: ax + by = c Factoring and using the Zero-Product Property Using the Quadratic Formula

Zero-Product Property Factor x2 + 7x + 10 = 0 Factors of 10 Sum of Factors 1 ∙ 10 11 2 ∙ 5 7 Both conditions must be true Factors of the problem are (x + 2)(x + 5) To find the real number solutions, use the Zero-Product Property Solve (x + 2)(x + 5) = 0 x + 2 = 0 or x + 5 = 0 x + 2 – 2 = 0 – 2 or x + 5 – 5 = 0 – 5 x = - 2 or x = - 5 There are two real number solutions to the quadratic equation.

Factor to Solve a Quadratic Equation Try It! Factor to Solve a Quadratic Equation Solve x2 + x – 42 = 0 by factoring. Factors of - 42 Sum of Factors 2 ∙ -21 -19 -6 ∙ 7 -6 + 7 = 1 Both conditions must be true Factors of the problem are (x + 7)(x -6) (x + 7)(x – 6) = 0 Factor using x2 + x – 42 x + 7 = 0 or x – 6 = 0 Use the Zero-Product Property. x = –7 or x = 6 Solve for x.

Practice 1. Use the zero property to solve (x – 3) (x – 7) = 0. 3 and 7 2. Use the zero property to solve – 3n(2n – 5) = 0. 0 and 2.5 3. Solve by factoring x2 – 16 x + 55 = 0. 5 and 11 4. Solve by factoring m2 – 5m – 14 = 0. -2 and 7