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Simplifying 0.5x2 + 1.5x + -54 = 0 Reorder the terms: -54 + 1.5x + 0.5x2 = 0 Solving -54 + 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'. -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 = 9 x = 9 Simplifying x = 9Subproblem 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 = -12 x = -12 Simplifying x = -12Solution
The solution to the problem is based on the solutions from the subproblems. x = {9, -12}
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