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Simplifying x(0.5x + 12) = 90 Reorder the terms: x(12 + 0.5x) = 90 (12 * x + 0.5x * x) = 90 (12x + 0.5x2) = 90 Solving 12x + 0.5x2 = 90 Solving for variable 'x'. Reorder the terms: -90 + 12x + 0.5x2 = 90 + -90 Combine like terms: 90 + -90 = 0 -90 + 12x + 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'. -180 + 24x + x2 = 0 Move the constant term to the right: Add '180' to each side of the equation. -180 + 24x + 180 + x2 = 0 + 180 Reorder the terms: -180 + 180 + 24x + x2 = 0 + 180 Combine like terms: -180 + 180 = 0 0 + 24x + x2 = 0 + 180 24x + x2 = 0 + 180 Combine like terms: 0 + 180 = 180 24x + x2 = 180 The x term is 24x. Take half its coefficient (12). Square it (144) and add it to both sides. Add '144' to each side of the equation. 24x + 144 + x2 = 180 + 144 Reorder the terms: 144 + 24x + x2 = 180 + 144 Combine like terms: 180 + 144 = 324 144 + 24x + x2 = 324 Factor a perfect square on the left side: (x + 12)(x + 12) = 324 Calculate the square root of the right side: 18 Break this problem into two subproblems by setting (x + 12) equal to 18 and -18.Subproblem 1
x + 12 = 18 Simplifying x + 12 = 18 Reorder the terms: 12 + x = 18 Solving 12 + x = 18 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-12' to each side of the equation. 12 + -12 + x = 18 + -12 Combine like terms: 12 + -12 = 0 0 + x = 18 + -12 x = 18 + -12 Combine like terms: 18 + -12 = 6 x = 6 Simplifying x = 6Subproblem 2
x + 12 = -18 Simplifying x + 12 = -18 Reorder the terms: 12 + x = -18 Solving 12 + x = -18 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-12' to each side of the equation. 12 + -12 + x = -18 + -12 Combine like terms: 12 + -12 = 0 0 + x = -18 + -12 x = -18 + -12 Combine like terms: -18 + -12 = -30 x = -30 Simplifying x = -30Solution
The solution to the problem is based on the solutions from the subproblems. x = {6, -30}
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