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Simplifying 32x2 + 32x + -48 = 0 Reorder the terms: -48 + 32x + 32x2 = 0 Solving -48 + 32x + 32x2 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '16'. 16(-3 + 2x + 2x2) = 0 Ignore the factor 16.Subproblem 1
Set the factor '(-3 + 2x + 2x2)' equal to zero and attempt to solve: Simplifying -3 + 2x + 2x2 = 0 Solving -3 + 2x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -1.5 + x + x2 = 0 Move the constant term to the right: Add '1.5' to each side of the equation. -1.5 + x + 1.5 + x2 = 0 + 1.5 Reorder the terms: -1.5 + 1.5 + x + x2 = 0 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x + x2 = 0 + 1.5 x + x2 = 0 + 1.5 Combine like terms: 0 + 1.5 = 1.5 x + x2 = 1.5 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. + 0.25 + x2 = 1.5 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + x2 = 1.5 + 0.25 Combine like terms: 1.5 + 0.25 = 1.75 1.25 + x2 = 1.75 Factor a perfect square on the left side: (x + 0.5)(x + 0.5) = 1.75 Calculate the square root of the right side: 1.322875656 Break this problem into two subproblems by setting (x + 0.5) equal to 1.322875656 and -1.322875656.Subproblem 1
x + 0.5 = 1.322875656 Simplifying x + 0.5 = 1.322875656 Reorder the terms: 0.5 + x = 1.322875656 Solving 0.5 + x = 1.322875656 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 = 1.322875656 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 1.322875656 + -0.5 x = 1.322875656 + -0.5 Combine like terms: 1.322875656 + -0.5 = 0.822875656 x = 0.822875656 Simplifying x = 0.822875656Subproblem 2
x + 0.5 = -1.322875656 Simplifying x + 0.5 = -1.322875656 Reorder the terms: 0.5 + x = -1.322875656 Solving 0.5 + x = -1.322875656 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 = -1.322875656 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -1.322875656 + -0.5 x = -1.322875656 + -0.5 Combine like terms: -1.322875656 + -0.5 = -1.822875656 x = -1.822875656 Simplifying x = -1.822875656Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.822875656, -1.822875656}Solution
x = {0.822875656, -1.822875656}
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