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Simplifying 8x2 = 24x + 68 Reorder the terms: 8x2 = 68 + 24x Solving 8x2 = 68 + 24x Solving for variable 'x'. Reorder the terms: -68 + -24x + 8x2 = 68 + 24x + -68 + -24x Reorder the terms: -68 + -24x + 8x2 = 68 + -68 + 24x + -24x Combine like terms: 68 + -68 = 0 -68 + -24x + 8x2 = 0 + 24x + -24x -68 + -24x + 8x2 = 24x + -24x Combine like terms: 24x + -24x = 0 -68 + -24x + 8x2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(-17 + -6x + 2x2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-17 + -6x + 2x2)' equal to zero and attempt to solve: Simplifying -17 + -6x + 2x2 = 0 Solving -17 + -6x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -8.5 + -3x + x2 = 0 Move the constant term to the right: Add '8.5' to each side of the equation. -8.5 + -3x + 8.5 + x2 = 0 + 8.5 Reorder the terms: -8.5 + 8.5 + -3x + x2 = 0 + 8.5 Combine like terms: -8.5 + 8.5 = 0.0 0.0 + -3x + x2 = 0 + 8.5 -3x + x2 = 0 + 8.5 Combine like terms: 0 + 8.5 = 8.5 -3x + x2 = 8.5 The x term is -3x. Take half its coefficient (-1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. -3x + 2.25 + x2 = 8.5 + 2.25 Reorder the terms: 2.25 + -3x + x2 = 8.5 + 2.25 Combine like terms: 8.5 + 2.25 = 10.75 2.25 + -3x + x2 = 10.75 Factor a perfect square on the left side: (x + -1.5)(x + -1.5) = 10.75 Calculate the square root of the right side: 3.278719262 Break this problem into two subproblems by setting (x + -1.5) equal to 3.278719262 and -3.278719262.Subproblem 1
x + -1.5 = 3.278719262 Simplifying x + -1.5 = 3.278719262 Reorder the terms: -1.5 + x = 3.278719262 Solving -1.5 + x = 3.278719262 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + x = 3.278719262 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = 3.278719262 + 1.5 x = 3.278719262 + 1.5 Combine like terms: 3.278719262 + 1.5 = 4.778719262 x = 4.778719262 Simplifying x = 4.778719262Subproblem 2
x + -1.5 = -3.278719262 Simplifying x + -1.5 = -3.278719262 Reorder the terms: -1.5 + x = -3.278719262 Solving -1.5 + x = -3.278719262 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + x = -3.278719262 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = -3.278719262 + 1.5 x = -3.278719262 + 1.5 Combine like terms: -3.278719262 + 1.5 = -1.778719262 x = -1.778719262 Simplifying x = -1.778719262Solution
The solution to the problem is based on the solutions from the subproblems. x = {4.778719262, -1.778719262}Solution
x = {4.778719262, -1.778719262}
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