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Simplifying -0.5(x + 3)(x + -4) = 0 Reorder the terms: -0.5(3 + x)(x + -4) = 0 Reorder the terms: -0.5(3 + x)(-4 + x) = 0 Multiply (3 + x) * (-4 + x) -0.5(3(-4 + x) + x(-4 + x)) = 0 -0.5((-4 * 3 + x * 3) + x(-4 + x)) = 0 -0.5((-12 + 3x) + x(-4 + x)) = 0 -0.5(-12 + 3x + (-4 * x + x * x)) = 0 -0.5(-12 + 3x + (-4x + x^{2})) = 0 Combine like terms: 3x + -4x = -1x -0.5(-12 + -1x + x^{2}) = 0 (-12 * -0.5 + -1x * -0.5 + x^{2}* -0.5) = 0 (6 + 0.5x + -0.5x^{2}) = 0 Solving 6 + 0.5x + -0.5x^{2}= 0 Solving for variable 'x'. Begin completing the square. Divide all terms by -0.5 the coefficient of the squared term: Divide each side by '-0.5'. -12 + -1x + x^{2}= 0 Move the constant term to the right: Add '12' to each side of the equation. -12 + -1x + 12 + x^{2}= 0 + 12 Reorder the terms: -12 + 12 + -1x + x^{2}= 0 + 12 Combine like terms: -12 + 12 = 0 0 + -1x + x^{2}= 0 + 12 -1x + x^{2}= 0 + 12 Combine like terms: 0 + 12 = 12 -1x + x^{2}= 12 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 + x^{2}= 12 + 0.25 Reorder the terms: 0.25 + -1x + x^{2}= 12 + 0.25 Combine like terms: 12 + 0.25 = 12.25 0.25 + -1x + x^{2}= 12.25 Factor a perfect square on the left side: (x + -0.5)(x + -0.5) = 12.25 Calculate the square root of the right side: 3.5 Break this problem into two subproblems by setting (x + -0.5) equal to 3.5 and -3.5.## Subproblem 1

x + -0.5 = 3.5 Simplifying x + -0.5 = 3.5 Reorder the terms: -0.5 + x = 3.5 Solving -0.5 + x = 3.5 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 = 3.5 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = 3.5 + 0.5 x = 3.5 + 0.5 Combine like terms: 3.5 + 0.5 = 4 x = 4 Simplifying x = 4## Subproblem 2

x + -0.5 = -3.5 Simplifying x + -0.5 = -3.5 Reorder the terms: -0.5 + x = -3.5 Solving -0.5 + x = -3.5 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 = -3.5 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = -3.5 + 0.5 x = -3.5 + 0.5 Combine like terms: -3.5 + 0.5 = -3 x = -3 Simplifying x = -3## Solution

The solution to the problem is based on the solutions from the subproblems. x = {4, -3}

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