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Simplifying 0 = (5x + -31)(x) + -75 Reorder the terms: 0 = (-31 + 5x)(x) + -75 Reorder the terms for easier multiplication: 0 = x(-31 + 5x) + -75 0 = (-31 * x + 5x * x) + -75 0 = (-31x + 5x2) + -75 Reorder the terms: 0 = -75 + -31x + 5x2 Solving 0 = -75 + -31x + 5x2 Solving for variable 'x'. Combine like terms: 0 + 75 = 75 75 + 31x + -5x2 = -75 + -31x + 5x2 + 75 + 31x + -5x2 Reorder the terms: 75 + 31x + -5x2 = -75 + 75 + -31x + 31x + 5x2 + -5x2 Combine like terms: -75 + 75 = 0 75 + 31x + -5x2 = 0 + -31x + 31x + 5x2 + -5x2 75 + 31x + -5x2 = -31x + 31x + 5x2 + -5x2 Combine like terms: -31x + 31x = 0 75 + 31x + -5x2 = 0 + 5x2 + -5x2 75 + 31x + -5x2 = 5x2 + -5x2 Combine like terms: 5x2 + -5x2 = 0 75 + 31x + -5x2 = 0 Begin completing the square. Divide all terms by -5 the coefficient of the squared term: Divide each side by '-5'. -15 + -6.2x + x2 = 0 Move the constant term to the right: Add '15' to each side of the equation. -15 + -6.2x + 15 + x2 = 0 + 15 Reorder the terms: -15 + 15 + -6.2x + x2 = 0 + 15 Combine like terms: -15 + 15 = 0 0 + -6.2x + x2 = 0 + 15 -6.2x + x2 = 0 + 15 Combine like terms: 0 + 15 = 15 -6.2x + x2 = 15 The x term is -6.2x. Take half its coefficient (-3.1). Square it (9.61) and add it to both sides. Add '9.61' to each side of the equation. -6.2x + 9.61 + x2 = 15 + 9.61 Reorder the terms: 9.61 + -6.2x + x2 = 15 + 9.61 Combine like terms: 15 + 9.61 = 24.61 9.61 + -6.2x + x2 = 24.61 Factor a perfect square on the left side: (x + -3.1)(x + -3.1) = 24.61 Calculate the square root of the right side: 4.960846702 Break this problem into two subproblems by setting (x + -3.1) equal to 4.960846702 and -4.960846702.Subproblem 1
x + -3.1 = 4.960846702 Simplifying x + -3.1 = 4.960846702 Reorder the terms: -3.1 + x = 4.960846702 Solving -3.1 + x = 4.960846702 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.1' to each side of the equation. -3.1 + 3.1 + x = 4.960846702 + 3.1 Combine like terms: -3.1 + 3.1 = 0.0 0.0 + x = 4.960846702 + 3.1 x = 4.960846702 + 3.1 Combine like terms: 4.960846702 + 3.1 = 8.060846702 x = 8.060846702 Simplifying x = 8.060846702Subproblem 2
x + -3.1 = -4.960846702 Simplifying x + -3.1 = -4.960846702 Reorder the terms: -3.1 + x = -4.960846702 Solving -3.1 + x = -4.960846702 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.1' to each side of the equation. -3.1 + 3.1 + x = -4.960846702 + 3.1 Combine like terms: -3.1 + 3.1 = 0.0 0.0 + x = -4.960846702 + 3.1 x = -4.960846702 + 3.1 Combine like terms: -4.960846702 + 3.1 = -1.860846702 x = -1.860846702 Simplifying x = -1.860846702Solution
The solution to the problem is based on the solutions from the subproblems. x = {8.060846702, -1.860846702}
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