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Simplifying 0 = -16x2 + 48x + 506 Reorder the terms: 0 = 506 + 48x + -16x2 Solving 0 = 506 + 48x + -16x2 Solving for variable 'x'. Combine like terms: 0 + -506 = -506 -506 + -48x + 16x2 = 506 + 48x + -16x2 + -506 + -48x + 16x2 Reorder the terms: -506 + -48x + 16x2 = 506 + -506 + 48x + -48x + -16x2 + 16x2 Combine like terms: 506 + -506 = 0 -506 + -48x + 16x2 = 0 + 48x + -48x + -16x2 + 16x2 -506 + -48x + 16x2 = 48x + -48x + -16x2 + 16x2 Combine like terms: 48x + -48x = 0 -506 + -48x + 16x2 = 0 + -16x2 + 16x2 -506 + -48x + 16x2 = -16x2 + 16x2 Combine like terms: -16x2 + 16x2 = 0 -506 + -48x + 16x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-253 + -24x + 8x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-253 + -24x + 8x2)' equal to zero and attempt to solve: Simplifying -253 + -24x + 8x2 = 0 Solving -253 + -24x + 8x2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -31.625 + -3x + x2 = 0 Move the constant term to the right: Add '31.625' to each side of the equation. -31.625 + -3x + 31.625 + x2 = 0 + 31.625 Reorder the terms: -31.625 + 31.625 + -3x + x2 = 0 + 31.625 Combine like terms: -31.625 + 31.625 = 0.000 0.000 + -3x + x2 = 0 + 31.625 -3x + x2 = 0 + 31.625 Combine like terms: 0 + 31.625 = 31.625 -3x + x2 = 31.625 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 = 31.625 + 2.25 Reorder the terms: 2.25 + -3x + x2 = 31.625 + 2.25 Combine like terms: 31.625 + 2.25 = 33.875 2.25 + -3x + x2 = 33.875 Factor a perfect square on the left side: (x + -1.5)(x + -1.5) = 33.875 Calculate the square root of the right side: 5.820223363 Break this problem into two subproblems by setting (x + -1.5) equal to 5.820223363 and -5.820223363.Subproblem 1
x + -1.5 = 5.820223363 Simplifying x + -1.5 = 5.820223363 Reorder the terms: -1.5 + x = 5.820223363 Solving -1.5 + x = 5.820223363 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 = 5.820223363 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = 5.820223363 + 1.5 x = 5.820223363 + 1.5 Combine like terms: 5.820223363 + 1.5 = 7.320223363 x = 7.320223363 Simplifying x = 7.320223363Subproblem 2
x + -1.5 = -5.820223363 Simplifying x + -1.5 = -5.820223363 Reorder the terms: -1.5 + x = -5.820223363 Solving -1.5 + x = -5.820223363 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 = -5.820223363 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = -5.820223363 + 1.5 x = -5.820223363 + 1.5 Combine like terms: -5.820223363 + 1.5 = -4.320223363 x = -4.320223363 Simplifying x = -4.320223363Solution
The solution to the problem is based on the solutions from the subproblems. x = {7.320223363, -4.320223363}Solution
x = {7.320223363, -4.320223363}
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