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Simplifying 0 = x2 + 31x + 53 Reorder the terms: 0 = 53 + 31x + x2 Solving 0 = 53 + 31x + x2 Solving for variable 'x'. Combine like terms: 0 + -53 = -53 -53 + -31x + -1x2 = 53 + 31x + x2 + -53 + -31x + -1x2 Reorder the terms: -53 + -31x + -1x2 = 53 + -53 + 31x + -31x + x2 + -1x2 Combine like terms: 53 + -53 = 0 -53 + -31x + -1x2 = 0 + 31x + -31x + x2 + -1x2 -53 + -31x + -1x2 = 31x + -31x + x2 + -1x2 Combine like terms: 31x + -31x = 0 -53 + -31x + -1x2 = 0 + x2 + -1x2 -53 + -31x + -1x2 = x2 + -1x2 Combine like terms: x2 + -1x2 = 0 -53 + -31x + -1x2 = 0 Factor out the Greatest Common Factor (GCF), '-1'. -1(53 + 31x + x2) = 0 Ignore the factor -1.Subproblem 1
Set the factor '(53 + 31x + x2)' equal to zero and attempt to solve: Simplifying 53 + 31x + x2 = 0 Solving 53 + 31x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '-53' to each side of the equation. 53 + 31x + -53 + x2 = 0 + -53 Reorder the terms: 53 + -53 + 31x + x2 = 0 + -53 Combine like terms: 53 + -53 = 0 0 + 31x + x2 = 0 + -53 31x + x2 = 0 + -53 Combine like terms: 0 + -53 = -53 31x + x2 = -53 The x term is 31x. Take half its coefficient (15.5). Square it (240.25) and add it to both sides. Add '240.25' to each side of the equation. 31x + 240.25 + x2 = -53 + 240.25 Reorder the terms: 240.25 + 31x + x2 = -53 + 240.25 Combine like terms: -53 + 240.25 = 187.25 240.25 + 31x + x2 = 187.25 Factor a perfect square on the left side: (x + 15.5)(x + 15.5) = 187.25 Calculate the square root of the right side: 13.683932183 Break this problem into two subproblems by setting (x + 15.5) equal to 13.683932183 and -13.683932183.Subproblem 1
x + 15.5 = 13.683932183 Simplifying x + 15.5 = 13.683932183 Reorder the terms: 15.5 + x = 13.683932183 Solving 15.5 + x = 13.683932183 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-15.5' to each side of the equation. 15.5 + -15.5 + x = 13.683932183 + -15.5 Combine like terms: 15.5 + -15.5 = 0.0 0.0 + x = 13.683932183 + -15.5 x = 13.683932183 + -15.5 Combine like terms: 13.683932183 + -15.5 = -1.816067817 x = -1.816067817 Simplifying x = -1.816067817Subproblem 2
x + 15.5 = -13.683932183 Simplifying x + 15.5 = -13.683932183 Reorder the terms: 15.5 + x = -13.683932183 Solving 15.5 + x = -13.683932183 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-15.5' to each side of the equation. 15.5 + -15.5 + x = -13.683932183 + -15.5 Combine like terms: 15.5 + -15.5 = 0.0 0.0 + x = -13.683932183 + -15.5 x = -13.683932183 + -15.5 Combine like terms: -13.683932183 + -15.5 = -29.183932183 x = -29.183932183 Simplifying x = -29.183932183Solution
The solution to the problem is based on the solutions from the subproblems. x = {-1.816067817, -29.183932183}Solution
x = {-1.816067817, -29.183932183}
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