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Simplifying 0 = -0.5 * 32x2 + 24x + 60 Multiply -0.5 * 32 0 = -16x2 + 24x + 60 Reorder the terms: 0 = 60 + 24x + -16x2 Solving 0 = 60 + 24x + -16x2 Solving for variable 'x'. Combine like terms: 0 + -60 = -60 -60 + -24x + 16x2 = 60 + 24x + -16x2 + -60 + -24x + 16x2 Reorder the terms: -60 + -24x + 16x2 = 60 + -60 + 24x + -24x + -16x2 + 16x2 Combine like terms: 60 + -60 = 0 -60 + -24x + 16x2 = 0 + 24x + -24x + -16x2 + 16x2 -60 + -24x + 16x2 = 24x + -24x + -16x2 + 16x2 Combine like terms: 24x + -24x = 0 -60 + -24x + 16x2 = 0 + -16x2 + 16x2 -60 + -24x + 16x2 = -16x2 + 16x2 Combine like terms: -16x2 + 16x2 = 0 -60 + -24x + 16x2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(-15 + -6x + 4x2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-15 + -6x + 4x2)' equal to zero and attempt to solve: Simplifying -15 + -6x + 4x2 = 0 Solving -15 + -6x + 4x2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -3.75 + -1.5x + x2 = 0 Move the constant term to the right: Add '3.75' to each side of the equation. -3.75 + -1.5x + 3.75 + x2 = 0 + 3.75 Reorder the terms: -3.75 + 3.75 + -1.5x + x2 = 0 + 3.75 Combine like terms: -3.75 + 3.75 = 0.00 0.00 + -1.5x + x2 = 0 + 3.75 -1.5x + x2 = 0 + 3.75 Combine like terms: 0 + 3.75 = 3.75 -1.5x + x2 = 3.75 The x term is -1.5x. Take half its coefficient (-0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. -1.5x + 0.5625 + x2 = 3.75 + 0.5625 Reorder the terms: 0.5625 + -1.5x + x2 = 3.75 + 0.5625 Combine like terms: 3.75 + 0.5625 = 4.3125 0.5625 + -1.5x + x2 = 4.3125 Factor a perfect square on the left side: (x + -0.75)(x + -0.75) = 4.3125 Calculate the square root of the right side: 2.076655966 Break this problem into two subproblems by setting (x + -0.75) equal to 2.076655966 and -2.076655966.Subproblem 1
x + -0.75 = 2.076655966 Simplifying x + -0.75 = 2.076655966 Reorder the terms: -0.75 + x = 2.076655966 Solving -0.75 + x = 2.076655966 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + x = 2.076655966 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + x = 2.076655966 + 0.75 x = 2.076655966 + 0.75 Combine like terms: 2.076655966 + 0.75 = 2.826655966 x = 2.826655966 Simplifying x = 2.826655966Subproblem 2
x + -0.75 = -2.076655966 Simplifying x + -0.75 = -2.076655966 Reorder the terms: -0.75 + x = -2.076655966 Solving -0.75 + x = -2.076655966 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + x = -2.076655966 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + x = -2.076655966 + 0.75 x = -2.076655966 + 0.75 Combine like terms: -2.076655966 + 0.75 = -1.326655966 x = -1.326655966 Simplifying x = -1.326655966Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.826655966, -1.326655966}Solution
x = {2.826655966, -1.326655966}
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