Solving Equations by Factoring Definition of Quadratic Equations Zero-Factor Property Strategy for Solving Quadratics

Standard Form Quadratic Equation Quadratic equations can be written in the form ax2 + bx + c = 0 where a, b, and c are real numbers with a  0. Standard form for a quadratic equation is in descending order equal to zero. BACK

Examples of Quadratic Equations Standard Form BACK

Zero-Factor Property If a and b are real numbers and if ab =0, then a = 0 or b = 0. BACK

Solve the equation (x + 2)(2x - 1)=0 By the zero factor property we know... Since the product is equal to zero then one of the factors must be zero. OR BACK

Solve the equation. Check your answers. Solution Set OR BACK

Solve each equation. Check your answers. OR Solution Set BACK

Solving a Quadratic Equation by Factoring Step 1 Write the equation in standard form. Step 2 Factor completely. Step 3 Use the zero-factor property. Set each factor with a variable equal to zero. Step 4 Solve each equation produced in step 3. BACK

Solve. BACK

Solve. BACK

Solve. BACK

Number Of Solutions The degree of a polynomial is equal to the number of solutions. Three solutions!!!

Set each of the three factors equal to 0. Example x (x + 1)(x – 3) = 0 Set each of the three factors equal to 0. x = 0 x + 1 = 0 x – 3 = 0 x = -1 x = 3 Solve the resulting equations. x = {0, -1, 3} Write the solution set. BACK

Solve. BACK

x2 – 9x + 20 = 0 (x – 4)(x – 5) = 0 x – 4 = 0 x = 4 x – 5 = 0 x = 5 Example: Standard form already Factor Set each factor = 0 Solve Write the solution set BACK

4x2 – 49 = 0 (2x + 7)(2x – 7) = 0 2x + 7 = 0 2x – 7 = 0 Example Standard form already Factor Set each factor = 0 Solve Write the solution set BACK

Solving Equations Now that we know how to factor, we can apply this knowledge to the solution of equations. How would you solve the following equation? x2 – 36 = 0 Step 1: Factor the polynomial. (x - 6)(x + 6) = 0

Solving Equations Step 2: Apply the zero product property which states that For all numbers a and b, if ab = 0, then a = 0, b = 0, or both a and b equal 0. (x - 6)(x + 6) = 0 Therefore (x – 6) = 0 or (x + 6) = 0. x + 6 = 0 - 6 -6 x – 6 = 0 + 6 +6 or x = 6 x = -6 This equation has two solutions or zeros: x = 6 or x = -6.

x2 – 25 = 0 x2 + 7x – 8 = 0 x2 – 12x + 36 = 0 c2 – 8c = 0 You Try It Solve the following equations. x2 – 25 = 0 x2 + 7x – 8 = 0 x2 – 12x + 36 = 0 c2 – 8c = 0

Summary of Steps Get a value of zero on one side of the equation. Factor the polynomial if possible. Apply the zero product property by setting each factor equal to zero. Solve for the variable.

Solving Equations by Factoring Definition of Quadratic Equations Zero-Factor Property Strategy for Solving Quadratics