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Simplifying 0 = 24x2 + -12x + -48 Reorder the terms: 0 = -48 + -12x + 24x2 Solving 0 = -48 + -12x + 24x2 Solving for variable 'x'. Combine like terms: 0 + 48 = 48 48 + 12x + -24x2 = -48 + -12x + 24x2 + 48 + 12x + -24x2 Reorder the terms: 48 + 12x + -24x2 = -48 + 48 + -12x + 12x + 24x2 + -24x2 Combine like terms: -48 + 48 = 0 48 + 12x + -24x2 = 0 + -12x + 12x + 24x2 + -24x2 48 + 12x + -24x2 = -12x + 12x + 24x2 + -24x2 Combine like terms: -12x + 12x = 0 48 + 12x + -24x2 = 0 + 24x2 + -24x2 48 + 12x + -24x2 = 24x2 + -24x2 Combine like terms: 24x2 + -24x2 = 0 48 + 12x + -24x2 = 0 Factor out the Greatest Common Factor (GCF), '12'. 12(4 + x + -2x2) = 0 Ignore the factor 12.Subproblem 1
Set the factor '(4 + x + -2x2)' equal to zero and attempt to solve: Simplifying 4 + x + -2x2 = 0 Solving 4 + x + -2x2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -2 + -0.5x + x2 = 0 Move the constant term to the right: Add '2' to each side of the equation. -2 + -0.5x + 2 + x2 = 0 + 2 Reorder the terms: -2 + 2 + -0.5x + x2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + -0.5x + x2 = 0 + 2 -0.5x + x2 = 0 + 2 Combine like terms: 0 + 2 = 2 -0.5x + x2 = 2 The x term is x. 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. -0.5x + 0.25 + x2 = 2 + 0.25 Reorder the terms: 0.25 + -0.5x + x2 = 2 + 0.25 Combine like terms: 2 + 0.25 = 2.25 0.25 + -0.5x + x2 = 2.25 Factor a perfect square on the left side: (x + 0.5)(x + 0.5) = 2.25 Calculate the square root of the right side: 1.5 Break this problem into two subproblems by setting (x + 0.5) equal to 1.5 and -1.5.Subproblem 1
x + 0.5 = 1.5 Simplifying x + 0.5 = 1.5 Reorder the terms: 0.5 + x = 1.5 Solving 0.5 + x = 1.5 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 = 1.5 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 1.5 + -0.5 x = 1.5 + -0.5 Combine like terms: 1.5 + -0.5 = 1 x = 1 Simplifying x = 1Subproblem 2
x + 0.5 = -1.5 Simplifying x + 0.5 = -1.5 Reorder the terms: 0.5 + x = -1.5 Solving 0.5 + x = -1.5 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 = -1.5 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -1.5 + -0.5 x = -1.5 + -0.5 Combine like terms: -1.5 + -0.5 = -2 x = -2 Simplifying x = -2Solution
The solution to the problem is based on the solutions from the subproblems. x = {1, -2}Solution
x = {1, -2}
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