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Simplifying 32 = -16x2 + 56x Reorder the terms: 32 = 56x + -16x2 Solving 32 = 56x + -16x2 Solving for variable 'x'. Reorder the terms: 32 + -56x + 16x2 = 56x + -56x + -16x2 + 16x2 Combine like terms: 56x + -56x = 0 32 + -56x + 16x2 = 0 + -16x2 + 16x2 32 + -56x + 16x2 = -16x2 + 16x2 Combine like terms: -16x2 + 16x2 = 0 32 + -56x + 16x2 = 0 Factor out the Greatest Common Factor (GCF), '8'. 8(4 + -7x + 2x2) = 0 Ignore the factor 8.Subproblem 1
Set the factor '(4 + -7x + 2x2)' equal to zero and attempt to solve: Simplifying 4 + -7x + 2x2 = 0 Solving 4 + -7x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 2 + -3.5x + x2 = 0 Move the constant term to the right: Add '-2' to each side of the equation. 2 + -3.5x + -2 + x2 = 0 + -2 Reorder the terms: 2 + -2 + -3.5x + x2 = 0 + -2 Combine like terms: 2 + -2 = 0 0 + -3.5x + x2 = 0 + -2 -3.5x + x2 = 0 + -2 Combine like terms: 0 + -2 = -2 -3.5x + x2 = -2 The x term is -3.5x. Take half its coefficient (-1.75). Square it (3.0625) and add it to both sides. Add '3.0625' to each side of the equation. -3.5x + 3.0625 + x2 = -2 + 3.0625 Reorder the terms: 3.0625 + -3.5x + x2 = -2 + 3.0625 Combine like terms: -2 + 3.0625 = 1.0625 3.0625 + -3.5x + x2 = 1.0625 Factor a perfect square on the left side: (x + -1.75)(x + -1.75) = 1.0625 Calculate the square root of the right side: 1.030776406 Break this problem into two subproblems by setting (x + -1.75) equal to 1.030776406 and -1.030776406.Subproblem 1
x + -1.75 = 1.030776406 Simplifying x + -1.75 = 1.030776406 Reorder the terms: -1.75 + x = 1.030776406 Solving -1.75 + x = 1.030776406 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.75' to each side of the equation. -1.75 + 1.75 + x = 1.030776406 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + x = 1.030776406 + 1.75 x = 1.030776406 + 1.75 Combine like terms: 1.030776406 + 1.75 = 2.780776406 x = 2.780776406 Simplifying x = 2.780776406Subproblem 2
x + -1.75 = -1.030776406 Simplifying x + -1.75 = -1.030776406 Reorder the terms: -1.75 + x = -1.030776406 Solving -1.75 + x = -1.030776406 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.75' to each side of the equation. -1.75 + 1.75 + x = -1.030776406 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + x = -1.030776406 + 1.75 x = -1.030776406 + 1.75 Combine like terms: -1.030776406 + 1.75 = 0.719223594 x = 0.719223594 Simplifying x = 0.719223594Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.780776406, 0.719223594}Solution
x = {2.780776406, 0.719223594}
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