135 1.4 Quadratic Equations Square half the coefficient of x: C 1 2 A b aB D 2 = A b 2aB 2 = b2 4a2 . x2 + b a x + b2 4a2 = - c a + b2 4a2 Add b2 4a2 to each side. (Step 3) ax + b 2ab 2 = b2 4a2 + -c a Factor. Use the commutative property. (Step 4) ax + b 2ab 2 = b2 4a2 + -4ac 4a2 Write fractions with a common denominator. ax + b 2ab 2 = b2 - 4ac 4a2 Add fractions. x + b 2a = {Bb2 - 4ac 4a2 Square root property (Step 5) x + b 2a = {2b2 - 4ac 2a Since a 70, 24a2 = 2a. x = -b 2a { 2b2 - 4ac 2a Subtract b 2a from each side. x = -b {2b2 - 4ac 2a Combine terms on the right. Quadratic Formula This result is also true for a 60. Quadratic Formula The solutions of the quadratic equation ax2 + bx + c = 0, where a≠0, are given by the quadratic formula. x = −b t!b2 −4ac 2a EXAMPLE 5 Using the Quadratic Formula (Real Solutions) Solve x2 - 4x = -2. SOLUTION x2 - 4x + 2 = 0 Write in standard form. Here a = 1, b = -4, and c = 2. x = -b {2b2 - 4ac 2a Quadratic formula x = -1-42 {21-422 - 4112122 2112 Substitute a = 1, b = -4, and c = 2. x = 4 {216 - 8 2 Simplify. x = 4 {222 2 216 - 8 = 28 = 24 # 2 = 222 x = 2 A 2{22 B 2 Factor out 2 in the numerator. x = 2{22 Lowest terms The solution set is E2{22 F . S Now Try Exercise 53. The fraction bar extends under -b. Factor first, then divide.
RkJQdWJsaXNoZXIy NjM5ODQ=