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Simplifying -2x2 + 2x + 112 = 38 Reorder the terms: 112 + 2x + -2x2 = 38 Solving 112 + 2x + -2x2 = 38 Solving for variable 'x'. Reorder the terms: 112 + -38 + 2x + -2x2 = 38 + -38 Combine like terms: 112 + -38 = 74 74 + 2x + -2x2 = 38 + -38 Combine like terms: 38 + -38 = 0 74 + 2x + -2x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(37 + x + -1x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(37 + x + -1x2)' equal to zero and attempt to solve: Simplifying 37 + x + -1x2 = 0 Solving 37 + x + -1x2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -37 + -1x + x2 = 0 Move the constant term to the right: Add '37' to each side of the equation. -37 + -1x + 37 + x2 = 0 + 37 Reorder the terms: -37 + 37 + -1x + x2 = 0 + 37 Combine like terms: -37 + 37 = 0 0 + -1x + x2 = 0 + 37 -1x + x2 = 0 + 37 Combine like terms: 0 + 37 = 37 -1x + x2 = 37 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. -1x + 0.25 + x2 = 37 + 0.25 Reorder the terms: 0.25 + -1x + x2 = 37 + 0.25 Combine like terms: 37 + 0.25 = 37.25 0.25 + -1x + x2 = 37.25 Factor a perfect square on the left side: (x + 0.5)(x + 0.5) = 37.25 Calculate the square root of the right side: 6.103277808 Break this problem into two subproblems by setting (x + 0.5) equal to 6.103277808 and -6.103277808.Subproblem 1
x + 0.5 = 6.103277808 Simplifying x + 0.5 = 6.103277808 Reorder the terms: 0.5 + x = 6.103277808 Solving 0.5 + x = 6.103277808 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 = 6.103277808 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 6.103277808 + -0.5 x = 6.103277808 + -0.5 Combine like terms: 6.103277808 + -0.5 = 5.603277808 x = 5.603277808 Simplifying x = 5.603277808Subproblem 2
x + 0.5 = -6.103277808 Simplifying x + 0.5 = -6.103277808 Reorder the terms: 0.5 + x = -6.103277808 Solving 0.5 + x = -6.103277808 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 = -6.103277808 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -6.103277808 + -0.5 x = -6.103277808 + -0.5 Combine like terms: -6.103277808 + -0.5 = -6.603277808 x = -6.603277808 Simplifying x = -6.603277808Solution
The solution to the problem is based on the solutions from the subproblems. x = {5.603277808, -6.603277808}Solution
x = {5.603277808, -6.603277808}
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