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. Examples of Quadratic Equations Standard Form p 81 18 p p 18 p 81 0 x 9 x 18 x 9 x 18 0 y 25 y 25 0 2 2 2 2 2 2 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. ( x 2) 0 x 2 1 x {2, } 2 OR (2 x 1) 0 2x 1 2x 1 2 2 1 x 2 Solve the equation. Check your answers. ( x 5)( x 2) 0 x 5 0 OR x 2 0 Solution Set x5 x 2 x {2, 5} Solve each equation. Check your answers. x(5 x 3) 0 x0 x0 Solution Set 3 x { , 0} 5 OR 5x 3 0 5 x 3 5 3 x 5 5 3 x 5 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. Solve. x 9 x 18 2 x 9 x 18 0 2 ( x 6)( x 3) 0 x {6, 3} Solve. m 3m 10 0 2 ( m 5)( m 2 ) 0 m {5, 2} Solve. x 7x 0 x ( x 7) 0 2 x0 x7 0 x {0, 7} Number Of Solutions • The degree of a polynomial is equal to the number of solutions. x 2 x 3x 3 2 Three solutions!!! x 2 x 3x 2 x( x 2 x 3) 3 2 x (x + 1)(x – 3) = 0 Set each of the three factors equal to 0. x=0 x+1=0 x = -1 x–3=0 x=3 Solve the resulting equations. Write the solution set. x = {0, -1, 3} Solve. r 2r 8 2 r 2r 8 0 ( r 2)( r 4 ) 0 r {2,4} 2 x2 – 9x + 20 = 0 (x – 4)(x – 5) = 0 x–4 = 0 x=4 x–5=0 x=5 x = {5, 4} • Standard form already • Factor • Set each factor = 0 • Solve • Write the solution set 4x2 – 49 = 0 Example (2x + 7)(2x – 7) = 0 • Standard form 2x + 7 = 0 already 7 x 2 2x – 7 = 0 7 x 2 • • • • Factor Set each factor = 0 Solve Write the solution set 7 7 x , 2 2 How would you solve the following equation? x2 – 36 = 0 Factor the polynomial. (x - 6)(x + 6) = 0 (x – 6) = 0 or (x + 6) = 0. Therefore x–6=0 + 6 +6 x=6 or x+6=0 - 6 -6 x = -6 This equation has two solutions or zeros: x = 6 or x = -6. Solve the following equations. 1. x2 – 25 = 0 x 5 x 5 0 5,5 2. x2 + 7x – 8 = 0 x 8 x 1 0 1,8 3. x2 – 12x + 36 = 0 x 6 x 6 0 6 4. c2 – 8c = 0 c ( c 8) 0 0,8 1. Get a value of zero on one side of the equation. 2. Factor the polynomial if possible. 3. Apply the zero product property by setting each factor equal to zero. 4. Solve for the variable. Solving Equations by Factoring Definition of Quadratic Equations Zero-Factor Property Strategy for Solving Quadratics

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# Section 5.6 Solving Quadratic Equations by Factoring