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Simplifying 27 = 0.5[2x + -3] * x Reorder the terms: 27 = 0.5[-3 + 2x] * x Reorder the terms for easier multiplication: 27 = 0.5x[-3 + 2x] 27 = [-3 * 0.5x + 2x * 0.5x] 27 = [-1.5x + 1x2] Solving 27 = -1.5x + 1x2 Solving for variable 'x'. Reorder the terms: 27 + 1.5x + -1x2 = -1.5x + 1.5x + 1x2 + -1x2 Combine like terms: -1.5x + 1.5x = 0.0 27 + 1.5x + -1x2 = 0.0 + 1x2 + -1x2 27 + 1.5x + -1x2 = 1x2 + -1x2 Combine like terms: 1x2 + -1x2 = 0 27 + 1.5x + -1x2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -27 + -1.5x + x2 = 0 Move the constant term to the right: Add '27' to each side of the equation. -27 + -1.5x + 27 + x2 = 0 + 27 Reorder the terms: -27 + 27 + -1.5x + x2 = 0 + 27 Combine like terms: -27 + 27 = 0 0 + -1.5x + x2 = 0 + 27 -1.5x + x2 = 0 + 27 Combine like terms: 0 + 27 = 27 -1.5x + x2 = 27 The x term is -1.5x. Take half its coefficient (-0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. -1.5x + 0.5625 + x2 = 27 + 0.5625 Reorder the terms: 0.5625 + -1.5x + x2 = 27 + 0.5625 Combine like terms: 27 + 0.5625 = 27.5625 0.5625 + -1.5x + x2 = 27.5625 Factor a perfect square on the left side: (x + -0.75)(x + -0.75) = 27.5625 Calculate the square root of the right side: 5.25 Break this problem into two subproblems by setting (x + -0.75) equal to 5.25 and -5.25.Subproblem 1
x + -0.75 = 5.25 Simplifying x + -0.75 = 5.25 Reorder the terms: -0.75 + x = 5.25 Solving -0.75 + x = 5.25 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + x = 5.25 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + x = 5.25 + 0.75 x = 5.25 + 0.75 Combine like terms: 5.25 + 0.75 = 6 x = 6 Simplifying x = 6Subproblem 2
x + -0.75 = -5.25 Simplifying x + -0.75 = -5.25 Reorder the terms: -0.75 + x = -5.25 Solving -0.75 + x = -5.25 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + x = -5.25 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + x = -5.25 + 0.75 x = -5.25 + 0.75 Combine like terms: -5.25 + 0.75 = -4.5 x = -4.5 Simplifying x = -4.5Solution
The solution to the problem is based on the solutions from the subproblems. x = {6, -4.5}
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