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Simplifying 0 = -15x2 + 240x + -640 Reorder the terms: 0 = -640 + 240x + -15x2 Solving 0 = -640 + 240x + -15x2 Solving for variable 'x'. Combine like terms: 0 + 640 = 640 640 + -240x + 15x2 = -640 + 240x + -15x2 + 640 + -240x + 15x2 Reorder the terms: 640 + -240x + 15x2 = -640 + 640 + 240x + -240x + -15x2 + 15x2 Combine like terms: -640 + 640 = 0 640 + -240x + 15x2 = 0 + 240x + -240x + -15x2 + 15x2 640 + -240x + 15x2 = 240x + -240x + -15x2 + 15x2 Combine like terms: 240x + -240x = 0 640 + -240x + 15x2 = 0 + -15x2 + 15x2 640 + -240x + 15x2 = -15x2 + 15x2 Combine like terms: -15x2 + 15x2 = 0 640 + -240x + 15x2 = 0 Factor out the Greatest Common Factor (GCF), '5'. 5(128 + -48x + 3x2) = 0 Ignore the factor 5.Subproblem 1
Set the factor '(128 + -48x + 3x2)' equal to zero and attempt to solve: Simplifying 128 + -48x + 3x2 = 0 Solving 128 + -48x + 3x2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 42.66666667 + -16x + x2 = 0 Move the constant term to the right: Add '-42.66666667' to each side of the equation. 42.66666667 + -16x + -42.66666667 + x2 = 0 + -42.66666667 Reorder the terms: 42.66666667 + -42.66666667 + -16x + x2 = 0 + -42.66666667 Combine like terms: 42.66666667 + -42.66666667 = 0.00000000 0.00000000 + -16x + x2 = 0 + -42.66666667 -16x + x2 = 0 + -42.66666667 Combine like terms: 0 + -42.66666667 = -42.66666667 -16x + x2 = -42.66666667 The x term is -16x. Take half its coefficient (-8). Square it (64) and add it to both sides. Add '64' to each side of the equation. -16x + 64 + x2 = -42.66666667 + 64 Reorder the terms: 64 + -16x + x2 = -42.66666667 + 64 Combine like terms: -42.66666667 + 64 = 21.33333333 64 + -16x + x2 = 21.33333333 Factor a perfect square on the left side: (x + -8)(x + -8) = 21.33333333 Calculate the square root of the right side: 4.618802153 Break this problem into two subproblems by setting (x + -8) equal to 4.618802153 and -4.618802153.Subproblem 1
x + -8 = 4.618802153 Simplifying x + -8 = 4.618802153 Reorder the terms: -8 + x = 4.618802153 Solving -8 + x = 4.618802153 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '8' to each side of the equation. -8 + 8 + x = 4.618802153 + 8 Combine like terms: -8 + 8 = 0 0 + x = 4.618802153 + 8 x = 4.618802153 + 8 Combine like terms: 4.618802153 + 8 = 12.618802153 x = 12.618802153 Simplifying x = 12.618802153Subproblem 2
x + -8 = -4.618802153 Simplifying x + -8 = -4.618802153 Reorder the terms: -8 + x = -4.618802153 Solving -8 + x = -4.618802153 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '8' to each side of the equation. -8 + 8 + x = -4.618802153 + 8 Combine like terms: -8 + 8 = 0 0 + x = -4.618802153 + 8 x = -4.618802153 + 8 Combine like terms: -4.618802153 + 8 = 3.381197847 x = 3.381197847 Simplifying x = 3.381197847Solution
The solution to the problem is based on the solutions from the subproblems. x = {12.618802153, 3.381197847}Solution
x = {12.618802153, 3.381197847}
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