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Simplifying -16x2 + 48x + 160 = 40 Reorder the terms: 160 + 48x + -16x2 = 40 Solving 160 + 48x + -16x2 = 40 Solving for variable 'x'. Reorder the terms: 160 + -40 + 48x + -16x2 = 40 + -40 Combine like terms: 160 + -40 = 120 120 + 48x + -16x2 = 40 + -40 Combine like terms: 40 + -40 = 0 120 + 48x + -16x2 = 0 Factor out the Greatest Common Factor (GCF), '8'. 8(15 + 6x + -2x2) = 0 Ignore the factor 8.Subproblem 1
Set the factor '(15 + 6x + -2x2)' equal to zero and attempt to solve: Simplifying 15 + 6x + -2x2 = 0 Solving 15 + 6x + -2x2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -7.5 + -3x + x2 = 0 Move the constant term to the right: Add '7.5' to each side of the equation. -7.5 + -3x + 7.5 + x2 = 0 + 7.5 Reorder the terms: -7.5 + 7.5 + -3x + x2 = 0 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + -3x + x2 = 0 + 7.5 -3x + x2 = 0 + 7.5 Combine like terms: 0 + 7.5 = 7.5 -3x + x2 = 7.5 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 = 7.5 + 2.25 Reorder the terms: 2.25 + -3x + x2 = 7.5 + 2.25 Combine like terms: 7.5 + 2.25 = 9.75 2.25 + -3x + x2 = 9.75 Factor a perfect square on the left side: (x + -1.5)(x + -1.5) = 9.75 Calculate the square root of the right side: 3.122498999 Break this problem into two subproblems by setting (x + -1.5) equal to 3.122498999 and -3.122498999.Subproblem 1
x + -1.5 = 3.122498999 Simplifying x + -1.5 = 3.122498999 Reorder the terms: -1.5 + x = 3.122498999 Solving -1.5 + x = 3.122498999 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 = 3.122498999 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = 3.122498999 + 1.5 x = 3.122498999 + 1.5 Combine like terms: 3.122498999 + 1.5 = 4.622498999 x = 4.622498999 Simplifying x = 4.622498999Subproblem 2
x + -1.5 = -3.122498999 Simplifying x + -1.5 = -3.122498999 Reorder the terms: -1.5 + x = -3.122498999 Solving -1.5 + x = -3.122498999 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 = -3.122498999 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = -3.122498999 + 1.5 x = -3.122498999 + 1.5 Combine like terms: -3.122498999 + 1.5 = -1.622498999 x = -1.622498999 Simplifying x = -1.622498999Solution
The solution to the problem is based on the solutions from the subproblems. x = {4.622498999, -1.622498999}Solution
x = {4.622498999, -1.622498999}
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