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Simplifying (0.5x2) + -1.5x + -27 = 0 Reorder the terms: -27 + -1.5x + (0.5x2) = 0 Solving -27 + -1.5x + (0.5x2) = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 0.5 the coefficient of the squared term: Divide each side by '0.5'. -54 + -3x + x2 = 0 Move the constant term to the right: Add '54' to each side of the equation. -54 + -3x + 54 + x2 = 0 + 54 Reorder the terms: -54 + 54 + -3x + x2 = 0 + 54 Combine like terms: -54 + 54 = 0 0 + -3x + x2 = 0 + 54 -3x + x2 = 0 + 54 Combine like terms: 0 + 54 = 54 -3x + x2 = 54 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 = 54 + 2.25 Reorder the terms: 2.25 + -3x + x2 = 54 + 2.25 Combine like terms: 54 + 2.25 = 56.25 2.25 + -3x + x2 = 56.25 Factor a perfect square on the left side: ((x) + -1.5)((x) + -1.5) = 56.25 Calculate the square root of the right side: 7.5 Break this problem into two subproblems by setting ((x) + -1.5) equal to 7.5 and -7.5.Subproblem 1
(x) + -1.5 = 7.5 Simplifying (x) + -1.5 = 7.5 x + -1.5 = 7.5 Reorder the terms: -1.5 + x = 7.5 Solving -1.5 + x = 7.5 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 = 7.5 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = 7.5 + 1.5 x = 7.5 + 1.5 Combine like terms: 7.5 + 1.5 = 9 x = 9 Simplifying x = 9Subproblem 2
(x) + -1.5 = -7.5 Simplifying (x) + -1.5 = -7.5 x + -1.5 = -7.5 Reorder the terms: -1.5 + x = -7.5 Solving -1.5 + x = -7.5 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 = -7.5 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = -7.5 + 1.5 x = -7.5 + 1.5 Combine like terms: -7.5 + 1.5 = -6 x = -6 Simplifying x = -6Solution
The solution to the problem is based on the solutions from the subproblems. x = {9, -6}
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