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Simplifying 42 = 0.5(x)(x + -5) Reorder the terms: 42 = 0.5x(-5 + x) 42 = (-5 * 0.5x + x * 0.5x) 42 = (-2.5x + 0.5x2) Solving 42 = -2.5x + 0.5x2 Solving for variable 'x'. Reorder the terms: 42 + 2.5x + -0.5x2 = -2.5x + 2.5x + 0.5x2 + -0.5x2 Combine like terms: -2.5x + 2.5x = 0.0 42 + 2.5x + -0.5x2 = 0.0 + 0.5x2 + -0.5x2 42 + 2.5x + -0.5x2 = 0.5x2 + -0.5x2 Combine like terms: 0.5x2 + -0.5x2 = 0.0 42 + 2.5x + -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'. -84 + -5x + x2 = 0 Move the constant term to the right: Add '84' to each side of the equation. -84 + -5x + 84 + x2 = 0 + 84 Reorder the terms: -84 + 84 + -5x + x2 = 0 + 84 Combine like terms: -84 + 84 = 0 0 + -5x + x2 = 0 + 84 -5x + x2 = 0 + 84 Combine like terms: 0 + 84 = 84 -5x + x2 = 84 The x term is -5x. Take half its coefficient (-2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. -5x + 6.25 + x2 = 84 + 6.25 Reorder the terms: 6.25 + -5x + x2 = 84 + 6.25 Combine like terms: 84 + 6.25 = 90.25 6.25 + -5x + x2 = 90.25 Factor a perfect square on the left side: (x + -2.5)(x + -2.5) = 90.25 Calculate the square root of the right side: 9.5 Break this problem into two subproblems by setting (x + -2.5) equal to 9.5 and -9.5.Subproblem 1
x + -2.5 = 9.5 Simplifying x + -2.5 = 9.5 Reorder the terms: -2.5 + x = 9.5 Solving -2.5 + x = 9.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + x = 9.5 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + x = 9.5 + 2.5 x = 9.5 + 2.5 Combine like terms: 9.5 + 2.5 = 12 x = 12 Simplifying x = 12Subproblem 2
x + -2.5 = -9.5 Simplifying x + -2.5 = -9.5 Reorder the terms: -2.5 + x = -9.5 Solving -2.5 + x = -9.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + x = -9.5 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + x = -9.5 + 2.5 x = -9.5 + 2.5 Combine like terms: -9.5 + 2.5 = -7 x = -7 Simplifying x = -7Solution
The solution to the problem is based on the solutions from the subproblems. x = {12, -7}
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