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Simplifying 90 = 0.5x(x + -3) Reorder the terms: 90 = 0.5x(-3 + x) 90 = (-3 * 0.5x + x * 0.5x) 90 = (-1.5x + 0.5x2) Solving 90 = -1.5x + 0.5x2 Solving for variable 'x'. Reorder the terms: 90 + 1.5x + -0.5x2 = -1.5x + 1.5x + 0.5x2 + -0.5x2 Combine like terms: -1.5x + 1.5x = 0.0 90 + 1.5x + -0.5x2 = 0.0 + 0.5x2 + -0.5x2 90 + 1.5x + -0.5x2 = 0.5x2 + -0.5x2 Combine like terms: 0.5x2 + -0.5x2 = 0.0 90 + 1.5x + -0.5x2 = 0.0 Begin completing the square. Divide all terms by -0.5 the coefficient of the squared term: Divide each side by '-0.5'. -180 + -3x + x2 = 0 Move the constant term to the right: Add '180' to each side of the equation. -180 + -3x + 180 + x2 = 0 + 180 Reorder the terms: -180 + 180 + -3x + x2 = 0 + 180 Combine like terms: -180 + 180 = 0 0 + -3x + x2 = 0 + 180 -3x + x2 = 0 + 180 Combine like terms: 0 + 180 = 180 -3x + x2 = 180 The x term is -3x. Take half its coefficient (-1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. -3x + 2.25 + x2 = 180 + 2.25 Reorder the terms: 2.25 + -3x + x2 = 180 + 2.25 Combine like terms: 180 + 2.25 = 182.25 2.25 + -3x + x2 = 182.25 Factor a perfect square on the left side: (x + -1.5)(x + -1.5) = 182.25 Calculate the square root of the right side: 13.5 Break this problem into two subproblems by setting (x + -1.5) equal to 13.5 and -13.5.Subproblem 1
x + -1.5 = 13.5 Simplifying x + -1.5 = 13.5 Reorder the terms: -1.5 + x = 13.5 Solving -1.5 + x = 13.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + x = 13.5 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = 13.5 + 1.5 x = 13.5 + 1.5 Combine like terms: 13.5 + 1.5 = 15 x = 15 Simplifying x = 15Subproblem 2
x + -1.5 = -13.5 Simplifying x + -1.5 = -13.5 Reorder the terms: -1.5 + x = -13.5 Solving -1.5 + x = -13.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + x = -13.5 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = -13.5 + 1.5 x = -13.5 + 1.5 Combine like terms: -13.5 + 1.5 = -12 x = -12 Simplifying x = -12Solution
The solution to the problem is based on the solutions from the subproblems. x = {15, -12}
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