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Simplifying 108 = -16x2 + -22x + 220 Reorder the terms: 108 = 220 + -22x + -16x2 Solving 108 = 220 + -22x + -16x2 Solving for variable 'x'. Combine like terms: 108 + -220 = -112 -112 + 22x + 16x2 = 220 + -22x + -16x2 + -220 + 22x + 16x2 Reorder the terms: -112 + 22x + 16x2 = 220 + -220 + -22x + 22x + -16x2 + 16x2 Combine like terms: 220 + -220 = 0 -112 + 22x + 16x2 = 0 + -22x + 22x + -16x2 + 16x2 -112 + 22x + 16x2 = -22x + 22x + -16x2 + 16x2 Combine like terms: -22x + 22x = 0 -112 + 22x + 16x2 = 0 + -16x2 + 16x2 -112 + 22x + 16x2 = -16x2 + 16x2 Combine like terms: -16x2 + 16x2 = 0 -112 + 22x + 16x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-56 + 11x + 8x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-56 + 11x + 8x2)' equal to zero and attempt to solve: Simplifying -56 + 11x + 8x2 = 0 Solving -56 + 11x + 8x2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -7 + 1.375x + x2 = 0 Move the constant term to the right: Add '7' to each side of the equation. -7 + 1.375x + 7 + x2 = 0 + 7 Reorder the terms: -7 + 7 + 1.375x + x2 = 0 + 7 Combine like terms: -7 + 7 = 0 0 + 1.375x + x2 = 0 + 7 1.375x + x2 = 0 + 7 Combine like terms: 0 + 7 = 7 1.375x + x2 = 7 The x term is 1.375x. Take half its coefficient (0.6875). Square it (0.47265625) and add it to both sides. Add '0.47265625' to each side of the equation. 1.375x + 0.47265625 + x2 = 7 + 0.47265625 Reorder the terms: 0.47265625 + 1.375x + x2 = 7 + 0.47265625 Combine like terms: 7 + 0.47265625 = 7.47265625 0.47265625 + 1.375x + x2 = 7.47265625 Factor a perfect square on the left side: (x + 0.6875)(x + 0.6875) = 7.47265625 Calculate the square root of the right side: 2.733615966 Break this problem into two subproblems by setting (x + 0.6875) equal to 2.733615966 and -2.733615966.Subproblem 1
x + 0.6875 = 2.733615966 Simplifying x + 0.6875 = 2.733615966 Reorder the terms: 0.6875 + x = 2.733615966 Solving 0.6875 + x = 2.733615966 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.6875' to each side of the equation. 0.6875 + -0.6875 + x = 2.733615966 + -0.6875 Combine like terms: 0.6875 + -0.6875 = 0.0000 0.0000 + x = 2.733615966 + -0.6875 x = 2.733615966 + -0.6875 Combine like terms: 2.733615966 + -0.6875 = 2.046115966 x = 2.046115966 Simplifying x = 2.046115966Subproblem 2
x + 0.6875 = -2.733615966 Simplifying x + 0.6875 = -2.733615966 Reorder the terms: 0.6875 + x = -2.733615966 Solving 0.6875 + x = -2.733615966 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.6875' to each side of the equation. 0.6875 + -0.6875 + x = -2.733615966 + -0.6875 Combine like terms: 0.6875 + -0.6875 = 0.0000 0.0000 + x = -2.733615966 + -0.6875 x = -2.733615966 + -0.6875 Combine like terms: -2.733615966 + -0.6875 = -3.421115966 x = -3.421115966 Simplifying x = -3.421115966Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.046115966, -3.421115966}Solution
x = {2.046115966, -3.421115966}
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