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Simplifying 110 = 2x2 + 14x + 24 Reorder the terms: 110 = 24 + 14x + 2x2 Solving 110 = 24 + 14x + 2x2 Solving for variable 'x'. Combine like terms: 110 + -24 = 86 86 + -14x + -2x2 = 24 + 14x + 2x2 + -24 + -14x + -2x2 Reorder the terms: 86 + -14x + -2x2 = 24 + -24 + 14x + -14x + 2x2 + -2x2 Combine like terms: 24 + -24 = 0 86 + -14x + -2x2 = 0 + 14x + -14x + 2x2 + -2x2 86 + -14x + -2x2 = 14x + -14x + 2x2 + -2x2 Combine like terms: 14x + -14x = 0 86 + -14x + -2x2 = 0 + 2x2 + -2x2 86 + -14x + -2x2 = 2x2 + -2x2 Combine like terms: 2x2 + -2x2 = 0 86 + -14x + -2x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(43 + -7x + -1x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(43 + -7x + -1x2)' equal to zero and attempt to solve: Simplifying 43 + -7x + -1x2 = 0 Solving 43 + -7x + -1x2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -43 + 7x + x2 = 0 Move the constant term to the right: Add '43' to each side of the equation. -43 + 7x + 43 + x2 = 0 + 43 Reorder the terms: -43 + 43 + 7x + x2 = 0 + 43 Combine like terms: -43 + 43 = 0 0 + 7x + x2 = 0 + 43 7x + x2 = 0 + 43 Combine like terms: 0 + 43 = 43 7x + x2 = 43 The x term is 7x. Take half its coefficient (3.5). Square it (12.25) and add it to both sides. Add '12.25' to each side of the equation. 7x + 12.25 + x2 = 43 + 12.25 Reorder the terms: 12.25 + 7x + x2 = 43 + 12.25 Combine like terms: 43 + 12.25 = 55.25 12.25 + 7x + x2 = 55.25 Factor a perfect square on the left side: (x + 3.5)(x + 3.5) = 55.25 Calculate the square root of the right side: 7.433034374 Break this problem into two subproblems by setting (x + 3.5) equal to 7.433034374 and -7.433034374.Subproblem 1
x + 3.5 = 7.433034374 Simplifying x + 3.5 = 7.433034374 Reorder the terms: 3.5 + x = 7.433034374 Solving 3.5 + x = 7.433034374 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + x = 7.433034374 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + x = 7.433034374 + -3.5 x = 7.433034374 + -3.5 Combine like terms: 7.433034374 + -3.5 = 3.933034374 x = 3.933034374 Simplifying x = 3.933034374Subproblem 2
x + 3.5 = -7.433034374 Simplifying x + 3.5 = -7.433034374 Reorder the terms: 3.5 + x = -7.433034374 Solving 3.5 + x = -7.433034374 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + x = -7.433034374 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + x = -7.433034374 + -3.5 x = -7.433034374 + -3.5 Combine like terms: -7.433034374 + -3.5 = -10.933034374 x = -10.933034374 Simplifying x = -10.933034374Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.933034374, -10.933034374}Solution
x = {3.933034374, -10.933034374}
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