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Simplifying 0.5x(x + -3) = 54 Reorder the terms: 0.5x(-3 + x) = 54 (-3 * 0.5x + x * 0.5x) = 54 (-1.5x + 0.5x2) = 54 Solving -1.5x + 0.5x2 = 54 Solving for variable 'x'. Reorder the terms: -54 + -1.5x + 0.5x2 = 54 + -54 Combine like terms: 54 + -54 = 0 -54 + -1.5x + 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'. -108 + -3x + x2 = 0 Move the constant term to the right: Add '108' to each side of the equation. -108 + -3x + 108 + x2 = 0 + 108 Reorder the terms: -108 + 108 + -3x + x2 = 0 + 108 Combine like terms: -108 + 108 = 0 0 + -3x + x2 = 0 + 108 -3x + x2 = 0 + 108 Combine like terms: 0 + 108 = 108 -3x + x2 = 108 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 = 108 + 2.25 Reorder the terms: 2.25 + -3x + x2 = 108 + 2.25 Combine like terms: 108 + 2.25 = 110.25 2.25 + -3x + x2 = 110.25 Factor a perfect square on the left side: (x + -1.5)(x + -1.5) = 110.25 Calculate the square root of the right side: 10.5 Break this problem into two subproblems by setting (x + -1.5) equal to 10.5 and -10.5.Subproblem 1
x + -1.5 = 10.5 Simplifying x + -1.5 = 10.5 Reorder the terms: -1.5 + x = 10.5 Solving -1.5 + x = 10.5 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 = 10.5 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = 10.5 + 1.5 x = 10.5 + 1.5 Combine like terms: 10.5 + 1.5 = 12 x = 12 Simplifying x = 12Subproblem 2
x + -1.5 = -10.5 Simplifying x + -1.5 = -10.5 Reorder the terms: -1.5 + x = -10.5 Solving -1.5 + x = -10.5 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 = -10.5 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = -10.5 + 1.5 x = -10.5 + 1.5 Combine like terms: -10.5 + 1.5 = -9 x = -9 Simplifying x = -9Solution
The solution to the problem is based on the solutions from the subproblems. x = {12, -9}
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