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Simplifying -0.5x2 + 1.5x + 40 = 5 Reorder the terms: 40 + 1.5x + -0.5x2 = 5 Solving 40 + 1.5x + -0.5x2 = 5 Solving for variable 'x'. Reorder the terms: 40 + -5 + 1.5x + -0.5x2 = 5 + -5 Combine like terms: 40 + -5 = 35 35 + 1.5x + -0.5x2 = 5 + -5 Combine like terms: 5 + -5 = 0 35 + 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'. -70 + -3x + x2 = 0 Move the constant term to the right: Add '70' to each side of the equation. -70 + -3x + 70 + x2 = 0 + 70 Reorder the terms: -70 + 70 + -3x + x2 = 0 + 70 Combine like terms: -70 + 70 = 0 0 + -3x + x2 = 0 + 70 -3x + x2 = 0 + 70 Combine like terms: 0 + 70 = 70 -3x + x2 = 70 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 = 70 + 2.25 Reorder the terms: 2.25 + -3x + x2 = 70 + 2.25 Combine like terms: 70 + 2.25 = 72.25 2.25 + -3x + x2 = 72.25 Factor a perfect square on the left side: (x + -1.5)(x + -1.5) = 72.25 Calculate the square root of the right side: 8.5 Break this problem into two subproblems by setting (x + -1.5) equal to 8.5 and -8.5.Subproblem 1
x + -1.5 = 8.5 Simplifying x + -1.5 = 8.5 Reorder the terms: -1.5 + x = 8.5 Solving -1.5 + x = 8.5 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 = 8.5 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = 8.5 + 1.5 x = 8.5 + 1.5 Combine like terms: 8.5 + 1.5 = 10 x = 10 Simplifying x = 10Subproblem 2
x + -1.5 = -8.5 Simplifying x + -1.5 = -8.5 Reorder the terms: -1.5 + x = -8.5 Solving -1.5 + x = -8.5 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 = -8.5 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = -8.5 + 1.5 x = -8.5 + 1.5 Combine like terms: -8.5 + 1.5 = -7 x = -7 Simplifying x = -7Solution
The solution to the problem is based on the solutions from the subproblems. x = {10, -7}
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