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Simplifying 32x2 + 2x + -240 = 0 Reorder the terms: -240 + 2x + 32x2 = 0 Solving -240 + 2x + 32x2 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '2'. 2(-120 + x + 16x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-120 + x + 16x2)' equal to zero and attempt to solve: Simplifying -120 + x + 16x2 = 0 Solving -120 + x + 16x2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -7.5 + 0.0625x + x2 = 0 Move the constant term to the right: Add '7.5' to each side of the equation. -7.5 + 0.0625x + 7.5 + x2 = 0 + 7.5 Reorder the terms: -7.5 + 7.5 + 0.0625x + x2 = 0 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + 0.0625x + x2 = 0 + 7.5 0.0625x + x2 = 0 + 7.5 Combine like terms: 0 + 7.5 = 7.5 0.0625x + x2 = 7.5 The x term is x. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. 0.0625x + 0.25 + x2 = 7.5 + 0.25 Reorder the terms: 0.25 + 0.0625x + x2 = 7.5 + 0.25 Combine like terms: 7.5 + 0.25 = 7.75 0.25 + 0.0625x + x2 = 7.75 Factor a perfect square on the left side: (x + 0.5)(x + 0.5) = 7.75 Calculate the square root of the right side: 2.783882181 Break this problem into two subproblems by setting (x + 0.5) equal to 2.783882181 and -2.783882181.Subproblem 1
x + 0.5 = 2.783882181 Simplifying x + 0.5 = 2.783882181 Reorder the terms: 0.5 + x = 2.783882181 Solving 0.5 + x = 2.783882181 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = 2.783882181 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 2.783882181 + -0.5 x = 2.783882181 + -0.5 Combine like terms: 2.783882181 + -0.5 = 2.283882181 x = 2.283882181 Simplifying x = 2.283882181Subproblem 2
x + 0.5 = -2.783882181 Simplifying x + 0.5 = -2.783882181 Reorder the terms: 0.5 + x = -2.783882181 Solving 0.5 + x = -2.783882181 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = -2.783882181 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -2.783882181 + -0.5 x = -2.783882181 + -0.5 Combine like terms: -2.783882181 + -0.5 = -3.283882181 x = -3.283882181 Simplifying x = -3.283882181Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.283882181, -3.283882181}Solution
x = {2.283882181, -3.283882181}
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