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Simplifying -0.5x2 + 10x = x + 16 Reorder the terms: 10x + -0.5x2 = x + 16 Reorder the terms: 10x + -0.5x2 = 16 + x Solving 10x + -0.5x2 = 16 + x Solving for variable 'x'. Reorder the terms: -16 + 10x + -1x + -0.5x2 = 16 + x + -16 + -1x Combine like terms: 10x + -1x = 9x -16 + 9x + -0.5x2 = 16 + x + -16 + -1x Reorder the terms: -16 + 9x + -0.5x2 = 16 + -16 + x + -1x Combine like terms: 16 + -16 = 0 -16 + 9x + -0.5x2 = 0 + x + -1x -16 + 9x + -0.5x2 = x + -1x Combine like terms: x + -1x = 0 -16 + 9x + -0.5x2 = 0 Begin completing the square. Divide all terms by -0.5 the coefficient of the squared term: Divide each side by '-0.5'. 32 + -18x + x2 = 0 Move the constant term to the right: Add '-32' to each side of the equation. 32 + -18x + -32 + x2 = 0 + -32 Reorder the terms: 32 + -32 + -18x + x2 = 0 + -32 Combine like terms: 32 + -32 = 0 0 + -18x + x2 = 0 + -32 -18x + x2 = 0 + -32 Combine like terms: 0 + -32 = -32 -18x + x2 = -32 The x term is -18x. Take half its coefficient (-9). Square it (81) and add it to both sides. Add '81' to each side of the equation. -18x + 81 + x2 = -32 + 81 Reorder the terms: 81 + -18x + x2 = -32 + 81 Combine like terms: -32 + 81 = 49 81 + -18x + x2 = 49 Factor a perfect square on the left side: (x + -9)(x + -9) = 49 Calculate the square root of the right side: 7 Break this problem into two subproblems by setting (x + -9) equal to 7 and -7.Subproblem 1
x + -9 = 7 Simplifying x + -9 = 7 Reorder the terms: -9 + x = 7 Solving -9 + x = 7 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '9' to each side of the equation. -9 + 9 + x = 7 + 9 Combine like terms: -9 + 9 = 0 0 + x = 7 + 9 x = 7 + 9 Combine like terms: 7 + 9 = 16 x = 16 Simplifying x = 16Subproblem 2
x + -9 = -7 Simplifying x + -9 = -7 Reorder the terms: -9 + x = -7 Solving -9 + x = -7 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '9' to each side of the equation. -9 + 9 + x = -7 + 9 Combine like terms: -9 + 9 = 0 0 + x = -7 + 9 x = -7 + 9 Combine like terms: -7 + 9 = 2 x = 2 Simplifying x = 2Solution
The solution to the problem is based on the solutions from the subproblems. x = {16, 2}
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