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Simplifying (13.5 + 0.75x) * x = -24 Reorder the terms for easier multiplication: x(13.5 + 0.75x) = -24 (13.5 * x + 0.75x * x) = -24 (13.5x + 0.75x2) = -24 Solving 13.5x + 0.75x2 = -24 Solving for variable 'x'. Reorder the terms: 24 + 13.5x + 0.75x2 = -24 + 24 Combine like terms: -24 + 24 = 0 24 + 13.5x + 0.75x2 = 0 Begin completing the square. Divide all terms by 0.75 the coefficient of the squared term: Divide each side by '0.75'. 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 = -2 x = -2 Simplifying x = -2Subproblem 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 = -16 x = -16 Simplifying x = -16Solution
The solution to the problem is based on the solutions from the subproblems. x = {-2, -16}
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