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Simplifying 0.5x(x + -12) + 2 = 1.25(x + 8) + -9.5 Reorder the terms: 0.5x(-12 + x) + 2 = 1.25(x + 8) + -9.5 (-12 * 0.5x + x * 0.5x) + 2 = 1.25(x + 8) + -9.5 (-6x + 0.5x2) + 2 = 1.25(x + 8) + -9.5 Reorder the terms: 2 + -6x + 0.5x2 = 1.25(x + 8) + -9.5 Reorder the terms: 2 + -6x + 0.5x2 = 1.25(8 + x) + -9.5 2 + -6x + 0.5x2 = (8 * 1.25 + x * 1.25) + -9.5 2 + -6x + 0.5x2 = (10 + 1.25x) + -9.5 Reorder the terms: 2 + -6x + 0.5x2 = 10 + -9.5 + 1.25x Combine like terms: 10 + -9.5 = 0.5 2 + -6x + 0.5x2 = 0.5 + 1.25x Solving 2 + -6x + 0.5x2 = 0.5 + 1.25x Solving for variable 'x'. Reorder the terms: 2 + -0.5 + -6x + -1.25x + 0.5x2 = 0.5 + 1.25x + -0.5 + -1.25x Combine like terms: 2 + -0.5 = 1.5 1.5 + -6x + -1.25x + 0.5x2 = 0.5 + 1.25x + -0.5 + -1.25x Combine like terms: -6x + -1.25x = -7.25x 1.5 + -7.25x + 0.5x2 = 0.5 + 1.25x + -0.5 + -1.25x Reorder the terms: 1.5 + -7.25x + 0.5x2 = 0.5 + -0.5 + 1.25x + -1.25x Combine like terms: 0.5 + -0.5 = 0.0 1.5 + -7.25x + 0.5x2 = 0.0 + 1.25x + -1.25x 1.5 + -7.25x + 0.5x2 = 1.25x + -1.25x Combine like terms: 1.25x + -1.25x = 0.00 1.5 + -7.25x + 0.5x2 = 0.00 Begin completing the square. Divide all terms by 0.5 the coefficient of the squared term: Divide each side by '0.5'. 3 + -14.5x + x2 = 0.0 Move the constant term to the right: Add '-3' to each side of the equation. 3 + -14.5x + -3 + x2 = 0.0 + -3 Reorder the terms: 3 + -3 + -14.5x + x2 = 0.0 + -3 Combine like terms: 3 + -3 = 0 0 + -14.5x + x2 = 0.0 + -3 -14.5x + x2 = 0.0 + -3 Combine like terms: 0.0 + -3 = -3 -14.5x + x2 = -3 The x term is -14.5x. Take half its coefficient (-7.25). Square it (52.5625) and add it to both sides. Add '52.5625' to each side of the equation. -14.5x + 52.5625 + x2 = -3 + 52.5625 Reorder the terms: 52.5625 + -14.5x + x2 = -3 + 52.5625 Combine like terms: -3 + 52.5625 = 49.5625 52.5625 + -14.5x + x2 = 49.5625 Factor a perfect square on the left side: (x + -7.25)(x + -7.25) = 49.5625 Calculate the square root of the right side: 7.04006392 Break this problem into two subproblems by setting (x + -7.25) equal to 7.04006392 and -7.04006392.Subproblem 1
x + -7.25 = 7.04006392 Simplifying x + -7.25 = 7.04006392 Reorder the terms: -7.25 + x = 7.04006392 Solving -7.25 + x = 7.04006392 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '7.25' to each side of the equation. -7.25 + 7.25 + x = 7.04006392 + 7.25 Combine like terms: -7.25 + 7.25 = 0.00 0.00 + x = 7.04006392 + 7.25 x = 7.04006392 + 7.25 Combine like terms: 7.04006392 + 7.25 = 14.29006392 x = 14.29006392 Simplifying x = 14.29006392Subproblem 2
x + -7.25 = -7.04006392 Simplifying x + -7.25 = -7.04006392 Reorder the terms: -7.25 + x = -7.04006392 Solving -7.25 + x = -7.04006392 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '7.25' to each side of the equation. -7.25 + 7.25 + x = -7.04006392 + 7.25 Combine like terms: -7.25 + 7.25 = 0.00 0.00 + x = -7.04006392 + 7.25 x = -7.04006392 + 7.25 Combine like terms: -7.04006392 + 7.25 = 0.20993608 x = 0.20993608 Simplifying x = 0.20993608Solution
The solution to the problem is based on the solutions from the subproblems. x = {14.29006392, 0.20993608}
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