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Simplifying -16x2 + 24x + 48 = 0 Reorder the terms: 48 + 24x + -16x2 = 0 Solving 48 + 24x + -16x2 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '8'. 8(6 + 3x + -2x2) = 0 Ignore the factor 8.Subproblem 1
Set the factor '(6 + 3x + -2x2)' equal to zero and attempt to solve: Simplifying 6 + 3x + -2x2 = 0 Solving 6 + 3x + -2x2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -3 + -1.5x + x2 = 0 Move the constant term to the right: Add '3' to each side of the equation. -3 + -1.5x + 3 + x2 = 0 + 3 Reorder the terms: -3 + 3 + -1.5x + x2 = 0 + 3 Combine like terms: -3 + 3 = 0 0 + -1.5x + x2 = 0 + 3 -1.5x + x2 = 0 + 3 Combine like terms: 0 + 3 = 3 -1.5x + x2 = 3 The x term is -1.5x. Take half its coefficient (-0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. -1.5x + 0.5625 + x2 = 3 + 0.5625 Reorder the terms: 0.5625 + -1.5x + x2 = 3 + 0.5625 Combine like terms: 3 + 0.5625 = 3.5625 0.5625 + -1.5x + x2 = 3.5625 Factor a perfect square on the left side: (x + -0.75)(x + -0.75) = 3.5625 Calculate the square root of the right side: 1.887458609 Break this problem into two subproblems by setting (x + -0.75) equal to 1.887458609 and -1.887458609.Subproblem 1
x + -0.75 = 1.887458609 Simplifying x + -0.75 = 1.887458609 Reorder the terms: -0.75 + x = 1.887458609 Solving -0.75 + x = 1.887458609 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + x = 1.887458609 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + x = 1.887458609 + 0.75 x = 1.887458609 + 0.75 Combine like terms: 1.887458609 + 0.75 = 2.637458609 x = 2.637458609 Simplifying x = 2.637458609Subproblem 2
x + -0.75 = -1.887458609 Simplifying x + -0.75 = -1.887458609 Reorder the terms: -0.75 + x = -1.887458609 Solving -0.75 + x = -1.887458609 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + x = -1.887458609 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + x = -1.887458609 + 0.75 x = -1.887458609 + 0.75 Combine like terms: -1.887458609 + 0.75 = -1.137458609 x = -1.137458609 Simplifying x = -1.137458609Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.637458609, -1.137458609}Solution
x = {2.637458609, -1.137458609}
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