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Simplifying 0.5(x + -2)(x + -7) = 18 Reorder the terms: 0.5(-2 + x)(x + -7) = 18 Reorder the terms: 0.5(-2 + x)(-7 + x) = 18 Multiply (-2 + x) * (-7 + x) 0.5(-2(-7 + x) + x(-7 + x)) = 18 0.5((-7 * -2 + x * -2) + x(-7 + x)) = 18 0.5((14 + -2x) + x(-7 + x)) = 18 0.5(14 + -2x + (-7 * x + x * x)) = 18 0.5(14 + -2x + (-7x + x2)) = 18 Combine like terms: -2x + -7x = -9x 0.5(14 + -9x + x2) = 18 (14 * 0.5 + -9x * 0.5 + x2 * 0.5) = 18 (7 + -4.5x + 0.5x2) = 18 Solving 7 + -4.5x + 0.5x2 = 18 Solving for variable 'x'. Reorder the terms: 7 + -18 + -4.5x + 0.5x2 = 18 + -18 Combine like terms: 7 + -18 = -11 -11 + -4.5x + 0.5x2 = 18 + -18 Combine like terms: 18 + -18 = 0 -11 + -4.5x + 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'. -22 + -9x + x2 = 0 Move the constant term to the right: Add '22' to each side of the equation. -22 + -9x + 22 + x2 = 0 + 22 Reorder the terms: -22 + 22 + -9x + x2 = 0 + 22 Combine like terms: -22 + 22 = 0 0 + -9x + x2 = 0 + 22 -9x + x2 = 0 + 22 Combine like terms: 0 + 22 = 22 -9x + x2 = 22 The x term is -9x. Take half its coefficient (-4.5). Square it (20.25) and add it to both sides. Add '20.25' to each side of the equation. -9x + 20.25 + x2 = 22 + 20.25 Reorder the terms: 20.25 + -9x + x2 = 22 + 20.25 Combine like terms: 22 + 20.25 = 42.25 20.25 + -9x + x2 = 42.25 Factor a perfect square on the left side: (x + -4.5)(x + -4.5) = 42.25 Calculate the square root of the right side: 6.5 Break this problem into two subproblems by setting (x + -4.5) equal to 6.5 and -6.5.Subproblem 1
x + -4.5 = 6.5 Simplifying x + -4.5 = 6.5 Reorder the terms: -4.5 + x = 6.5 Solving -4.5 + x = 6.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + x = 6.5 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + x = 6.5 + 4.5 x = 6.5 + 4.5 Combine like terms: 6.5 + 4.5 = 11 x = 11 Simplifying x = 11Subproblem 2
x + -4.5 = -6.5 Simplifying x + -4.5 = -6.5 Reorder the terms: -4.5 + x = -6.5 Solving -4.5 + x = -6.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + x = -6.5 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + x = -6.5 + 4.5 x = -6.5 + 4.5 Combine like terms: -6.5 + 4.5 = -2 x = -2 Simplifying x = -2Solution
The solution to the problem is based on the solutions from the subproblems. x = {11, -2}
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