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Simplifying 20 = -16x2 + 48x + 4 Reorder the terms: 20 = 4 + 48x + -16x2 Solving 20 = 4 + 48x + -16x2 Solving for variable 'x'. Combine like terms: 20 + -4 = 16 16 + -48x + 16x2 = 4 + 48x + -16x2 + -4 + -48x + 16x2 Reorder the terms: 16 + -48x + 16x2 = 4 + -4 + 48x + -48x + -16x2 + 16x2 Combine like terms: 4 + -4 = 0 16 + -48x + 16x2 = 0 + 48x + -48x + -16x2 + 16x2 16 + -48x + 16x2 = 48x + -48x + -16x2 + 16x2 Combine like terms: 48x + -48x = 0 16 + -48x + 16x2 = 0 + -16x2 + 16x2 16 + -48x + 16x2 = -16x2 + 16x2 Combine like terms: -16x2 + 16x2 = 0 16 + -48x + 16x2 = 0 Factor out the Greatest Common Factor (GCF), '16'. 16(1 + -3x + x2) = 0 Ignore the factor 16.Subproblem 1
Set the factor '(1 + -3x + x2)' equal to zero and attempt to solve: Simplifying 1 + -3x + x2 = 0 Solving 1 + -3x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '-1' to each side of the equation. 1 + -3x + -1 + x2 = 0 + -1 Reorder the terms: 1 + -1 + -3x + x2 = 0 + -1 Combine like terms: 1 + -1 = 0 0 + -3x + x2 = 0 + -1 -3x + x2 = 0 + -1 Combine like terms: 0 + -1 = -1 -3x + x2 = -1 The x term is -3x. Take half its coefficient (-1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. -3x + 2.25 + x2 = -1 + 2.25 Reorder the terms: 2.25 + -3x + x2 = -1 + 2.25 Combine like terms: -1 + 2.25 = 1.25 2.25 + -3x + x2 = 1.25 Factor a perfect square on the left side: (x + -1.5)(x + -1.5) = 1.25 Calculate the square root of the right side: 1.118033989 Break this problem into two subproblems by setting (x + -1.5) equal to 1.118033989 and -1.118033989.Subproblem 1
x + -1.5 = 1.118033989 Simplifying x + -1.5 = 1.118033989 Reorder the terms: -1.5 + x = 1.118033989 Solving -1.5 + x = 1.118033989 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + x = 1.118033989 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = 1.118033989 + 1.5 x = 1.118033989 + 1.5 Combine like terms: 1.118033989 + 1.5 = 2.618033989 x = 2.618033989 Simplifying x = 2.618033989Subproblem 2
x + -1.5 = -1.118033989 Simplifying x + -1.5 = -1.118033989 Reorder the terms: -1.5 + x = -1.118033989 Solving -1.5 + x = -1.118033989 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + x = -1.118033989 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = -1.118033989 + 1.5 x = -1.118033989 + 1.5 Combine like terms: -1.118033989 + 1.5 = 0.381966011 x = 0.381966011 Simplifying x = 0.381966011Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.618033989, 0.381966011}Solution
x = {2.618033989, 0.381966011}
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