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Simplifying 0.5x(x + -3) = 65 Reorder the terms: 0.5x(-3 + x) = 65 (-3 * 0.5x + x * 0.5x) = 65 (-1.5x + 0.5x2) = 65 Solving -1.5x + 0.5x2 = 65 Solving for variable 'x'. Reorder the terms: -65 + -1.5x + 0.5x2 = 65 + -65 Combine like terms: 65 + -65 = 0 -65 + -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'. -130 + -3x + x2 = 0 Move the constant term to the right: Add '130' to each side of the equation. -130 + -3x + 130 + x2 = 0 + 130 Reorder the terms: -130 + 130 + -3x + x2 = 0 + 130 Combine like terms: -130 + 130 = 0 0 + -3x + x2 = 0 + 130 -3x + x2 = 0 + 130 Combine like terms: 0 + 130 = 130 -3x + x2 = 130 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 = 130 + 2.25 Reorder the terms: 2.25 + -3x + x2 = 130 + 2.25 Combine like terms: 130 + 2.25 = 132.25 2.25 + -3x + x2 = 132.25 Factor a perfect square on the left side: (x + -1.5)(x + -1.5) = 132.25 Calculate the square root of the right side: 11.5 Break this problem into two subproblems by setting (x + -1.5) equal to 11.5 and -11.5.Subproblem 1
x + -1.5 = 11.5 Simplifying x + -1.5 = 11.5 Reorder the terms: -1.5 + x = 11.5 Solving -1.5 + x = 11.5 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 = 11.5 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = 11.5 + 1.5 x = 11.5 + 1.5 Combine like terms: 11.5 + 1.5 = 13 x = 13 Simplifying x = 13Subproblem 2
x + -1.5 = -11.5 Simplifying x + -1.5 = -11.5 Reorder the terms: -1.5 + x = -11.5 Solving -1.5 + x = -11.5 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 = -11.5 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = -11.5 + 1.5 x = -11.5 + 1.5 Combine like terms: -11.5 + 1.5 = -10 x = -10 Simplifying x = -10Solution
The solution to the problem is based on the solutions from the subproblems. x = {13, -10}
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