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Simplifying 130 = ((0.5)(x + -7))(x) Reorder the terms: 130 = (0.5(-7 + x))(x) 130 = ((-7 * 0.5 + x * 0.5))(x) 130 = ((-3.5 + 0.5x))(x) Reorder the terms for easier multiplication: 130 = x(-3.5 + 0.5x) 130 = (-3.5 * x + 0.5x * x) 130 = (-3.5x + 0.5x2) Solving 130 = -3.5x + 0.5x2 Solving for variable 'x'. Reorder the terms: 130 + 3.5x + -0.5x2 = -3.5x + 3.5x + 0.5x2 + -0.5x2 Combine like terms: -3.5x + 3.5x = 0.0 130 + 3.5x + -0.5x2 = 0.0 + 0.5x2 + -0.5x2 130 + 3.5x + -0.5x2 = 0.5x2 + -0.5x2 Combine like terms: 0.5x2 + -0.5x2 = 0.0 130 + 3.5x + -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'. -260 + -7x + x2 = 0 Move the constant term to the right: Add '260' to each side of the equation. -260 + -7x + 260 + x2 = 0 + 260 Reorder the terms: -260 + 260 + -7x + x2 = 0 + 260 Combine like terms: -260 + 260 = 0 0 + -7x + x2 = 0 + 260 -7x + x2 = 0 + 260 Combine like terms: 0 + 260 = 260 -7x + x2 = 260 The x term is -7x. Take half its coefficient (-3.5). Square it (12.25) and add it to both sides. Add '12.25' to each side of the equation. -7x + 12.25 + x2 = 260 + 12.25 Reorder the terms: 12.25 + -7x + x2 = 260 + 12.25 Combine like terms: 260 + 12.25 = 272.25 12.25 + -7x + x2 = 272.25 Factor a perfect square on the left side: (x + -3.5)(x + -3.5) = 272.25 Calculate the square root of the right side: 16.5 Break this problem into two subproblems by setting (x + -3.5) equal to 16.5 and -16.5.Subproblem 1
x + -3.5 = 16.5 Simplifying x + -3.5 = 16.5 Reorder the terms: -3.5 + x = 16.5 Solving -3.5 + x = 16.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + x = 16.5 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + x = 16.5 + 3.5 x = 16.5 + 3.5 Combine like terms: 16.5 + 3.5 = 20 x = 20 Simplifying x = 20Subproblem 2
x + -3.5 = -16.5 Simplifying x + -3.5 = -16.5 Reorder the terms: -3.5 + x = -16.5 Solving -3.5 + x = -16.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + x = -16.5 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + x = -16.5 + 3.5 x = -16.5 + 3.5 Combine like terms: -16.5 + 3.5 = -13 x = -13 Simplifying x = -13Solution
The solution to the problem is based on the solutions from the subproblems. x = {20, -13}
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