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Simplifying 13 = 0.5(x + 5)(x + -6) Reorder the terms: 13 = 0.5(5 + x)(x + -6) Reorder the terms: 13 = 0.5(5 + x)(-6 + x) Multiply (5 + x) * (-6 + x) 13 = 0.5(5(-6 + x) + x(-6 + x)) 13 = 0.5((-6 * 5 + x * 5) + x(-6 + x)) 13 = 0.5((-30 + 5x) + x(-6 + x)) 13 = 0.5(-30 + 5x + (-6 * x + x * x)) 13 = 0.5(-30 + 5x + (-6x + x2)) Combine like terms: 5x + -6x = -1x 13 = 0.5(-30 + -1x + x2) 13 = (-30 * 0.5 + -1x * 0.5 + x2 * 0.5) 13 = (-15 + -0.5x + 0.5x2) Solving 13 = -15 + -0.5x + 0.5x2 Solving for variable 'x'. Combine like terms: 13 + 15 = 28 28 + 0.5x + -0.5x2 = -15 + -0.5x + 0.5x2 + 15 + 0.5x + -0.5x2 Reorder the terms: 28 + 0.5x + -0.5x2 = -15 + 15 + -0.5x + 0.5x + 0.5x2 + -0.5x2 Combine like terms: -15 + 15 = 0 28 + 0.5x + -0.5x2 = 0 + -0.5x + 0.5x + 0.5x2 + -0.5x2 28 + 0.5x + -0.5x2 = -0.5x + 0.5x + 0.5x2 + -0.5x2 Combine like terms: -0.5x + 0.5x = 0.0 28 + 0.5x + -0.5x2 = 0.0 + 0.5x2 + -0.5x2 28 + 0.5x + -0.5x2 = 0.5x2 + -0.5x2 Combine like terms: 0.5x2 + -0.5x2 = 0.0 28 + 0.5x + -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'. -56 + -1x + x2 = 0 Move the constant term to the right: Add '56' to each side of the equation. -56 + -1x + 56 + x2 = 0 + 56 Reorder the terms: -56 + 56 + -1x + x2 = 0 + 56 Combine like terms: -56 + 56 = 0 0 + -1x + x2 = 0 + 56 -1x + x2 = 0 + 56 Combine like terms: 0 + 56 = 56 -1x + x2 = 56 The x term is -1x. Take half its coefficient (-0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -1x + 0.25 + x2 = 56 + 0.25 Reorder the terms: 0.25 + -1x + x2 = 56 + 0.25 Combine like terms: 56 + 0.25 = 56.25 0.25 + -1x + x2 = 56.25 Factor a perfect square on the left side: (x + -0.5)(x + -0.5) = 56.25 Calculate the square root of the right side: 7.5 Break this problem into two subproblems by setting (x + -0.5) equal to 7.5 and -7.5.Subproblem 1
x + -0.5 = 7.5 Simplifying x + -0.5 = 7.5 Reorder the terms: -0.5 + x = 7.5 Solving -0.5 + x = 7.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + x = 7.5 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = 7.5 + 0.5 x = 7.5 + 0.5 Combine like terms: 7.5 + 0.5 = 8 x = 8 Simplifying x = 8Subproblem 2
x + -0.5 = -7.5 Simplifying x + -0.5 = -7.5 Reorder the terms: -0.5 + x = -7.5 Solving -0.5 + x = -7.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + x = -7.5 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = -7.5 + 0.5 x = -7.5 + 0.5 Combine like terms: -7.5 + 0.5 = -7 x = -7 Simplifying x = -7Solution
The solution to the problem is based on the solutions from the subproblems. x = {8, -7}
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