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Simplifying 12 = 0.5(x)(x + -5) Reorder the terms: 12 = 0.5x(-5 + x) 12 = (-5 * 0.5x + x * 0.5x) 12 = (-2.5x + 0.5x2) Solving 12 = -2.5x + 0.5x2 Solving for variable 'x'. Reorder the terms: 12 + 2.5x + -0.5x2 = -2.5x + 2.5x + 0.5x2 + -0.5x2 Combine like terms: -2.5x + 2.5x = 0.0 12 + 2.5x + -0.5x2 = 0.0 + 0.5x2 + -0.5x2 12 + 2.5x + -0.5x2 = 0.5x2 + -0.5x2 Combine like terms: 0.5x2 + -0.5x2 = 0.0 12 + 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'. -24 + -5x + x2 = 0 Move the constant term to the right: Add '24' to each side of the equation. -24 + -5x + 24 + x2 = 0 + 24 Reorder the terms: -24 + 24 + -5x + x2 = 0 + 24 Combine like terms: -24 + 24 = 0 0 + -5x + x2 = 0 + 24 -5x + x2 = 0 + 24 Combine like terms: 0 + 24 = 24 -5x + x2 = 24 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 = 24 + 6.25 Reorder the terms: 6.25 + -5x + x2 = 24 + 6.25 Combine like terms: 24 + 6.25 = 30.25 6.25 + -5x + x2 = 30.25 Factor a perfect square on the left side: (x + -2.5)(x + -2.5) = 30.25 Calculate the square root of the right side: 5.5 Break this problem into two subproblems by setting (x + -2.5) equal to 5.5 and -5.5.Subproblem 1
x + -2.5 = 5.5 Simplifying x + -2.5 = 5.5 Reorder the terms: -2.5 + x = 5.5 Solving -2.5 + x = 5.5 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 = 5.5 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + x = 5.5 + 2.5 x = 5.5 + 2.5 Combine like terms: 5.5 + 2.5 = 8 x = 8 Simplifying x = 8Subproblem 2
x + -2.5 = -5.5 Simplifying x + -2.5 = -5.5 Reorder the terms: -2.5 + x = -5.5 Solving -2.5 + x = -5.5 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 = -5.5 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + x = -5.5 + 2.5 x = -5.5 + 2.5 Combine like terms: -5.5 + 2.5 = -3 x = -3 Simplifying x = -3Solution
The solution to the problem is based on the solutions from the subproblems. x = {8, -3}
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