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Simplifying 30 = 0.5(x2 + 17x) Reorder the terms: 30 = 0.5(17x + x2) 30 = (17x * 0.5 + x2 * 0.5) 30 = (8.5x + 0.5x2) Solving 30 = 8.5x + 0.5x2 Solving for variable 'x'. Reorder the terms: 30 + -8.5x + -0.5x2 = 8.5x + -8.5x + 0.5x2 + -0.5x2 Combine like terms: 8.5x + -8.5x = 0.0 30 + -8.5x + -0.5x2 = 0.0 + 0.5x2 + -0.5x2 30 + -8.5x + -0.5x2 = 0.5x2 + -0.5x2 Combine like terms: 0.5x2 + -0.5x2 = 0.0 30 + -8.5x + -0.5x2 = 0.0 Begin completing the square. Divide all terms by -0.5 the coefficient of the squared term: Divide each side by '-0.5'. -60 + 17x + x2 = 0 Move the constant term to the right: Add '60' to each side of the equation. -60 + 17x + 60 + x2 = 0 + 60 Reorder the terms: -60 + 60 + 17x + x2 = 0 + 60 Combine like terms: -60 + 60 = 0 0 + 17x + x2 = 0 + 60 17x + x2 = 0 + 60 Combine like terms: 0 + 60 = 60 17x + x2 = 60 The x term is 17x. Take half its coefficient (8.5). Square it (72.25) and add it to both sides. Add '72.25' to each side of the equation. 17x + 72.25 + x2 = 60 + 72.25 Reorder the terms: 72.25 + 17x + x2 = 60 + 72.25 Combine like terms: 60 + 72.25 = 132.25 72.25 + 17x + x2 = 132.25 Factor a perfect square on the left side: (x + 8.5)(x + 8.5) = 132.25 Calculate the square root of the right side: 11.5 Break this problem into two subproblems by setting (x + 8.5) equal to 11.5 and -11.5.Subproblem 1
x + 8.5 = 11.5 Simplifying x + 8.5 = 11.5 Reorder the terms: 8.5 + x = 11.5 Solving 8.5 + x = 11.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-8.5' to each side of the equation. 8.5 + -8.5 + x = 11.5 + -8.5 Combine like terms: 8.5 + -8.5 = 0.0 0.0 + x = 11.5 + -8.5 x = 11.5 + -8.5 Combine like terms: 11.5 + -8.5 = 3 x = 3 Simplifying x = 3Subproblem 2
x + 8.5 = -11.5 Simplifying x + 8.5 = -11.5 Reorder the terms: 8.5 + x = -11.5 Solving 8.5 + x = -11.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-8.5' to each side of the equation. 8.5 + -8.5 + x = -11.5 + -8.5 Combine like terms: 8.5 + -8.5 = 0.0 0.0 + x = -11.5 + -8.5 x = -11.5 + -8.5 Combine like terms: -11.5 + -8.5 = -20 x = -20 Simplifying x = -20Solution
The solution to the problem is based on the solutions from the subproblems. x = {3, -20}
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