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Simplifying (3x + 50)(6x + -10) = 180 Reorder the terms: (50 + 3x)(6x + -10) = 180 Reorder the terms: (50 + 3x)(-10 + 6x) = 180 Multiply (50 + 3x) * (-10 + 6x) (50(-10 + 6x) + 3x * (-10 + 6x)) = 180 ((-10 * 50 + 6x * 50) + 3x * (-10 + 6x)) = 180 ((-500 + 300x) + 3x * (-10 + 6x)) = 180 (-500 + 300x + (-10 * 3x + 6x * 3x)) = 180 (-500 + 300x + (-30x + 18x2)) = 180 Combine like terms: 300x + -30x = 270x (-500 + 270x + 18x2) = 180 Solving -500 + 270x + 18x2 = 180 Solving for variable 'x'. Reorder the terms: -500 + -180 + 270x + 18x2 = 180 + -180 Combine like terms: -500 + -180 = -680 -680 + 270x + 18x2 = 180 + -180 Combine like terms: 180 + -180 = 0 -680 + 270x + 18x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-340 + 135x + 9x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-340 + 135x + 9x2)' equal to zero and attempt to solve: Simplifying -340 + 135x + 9x2 = 0 Solving -340 + 135x + 9x2 = 0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -37.77777778 + 15x + x2 = 0 Move the constant term to the right: Add '37.77777778' to each side of the equation. -37.77777778 + 15x + 37.77777778 + x2 = 0 + 37.77777778 Reorder the terms: -37.77777778 + 37.77777778 + 15x + x2 = 0 + 37.77777778 Combine like terms: -37.77777778 + 37.77777778 = 0.00000000 0.00000000 + 15x + x2 = 0 + 37.77777778 15x + x2 = 0 + 37.77777778 Combine like terms: 0 + 37.77777778 = 37.77777778 15x + x2 = 37.77777778 The x term is 15x. Take half its coefficient (7.5). Square it (56.25) and add it to both sides. Add '56.25' to each side of the equation. 15x + 56.25 + x2 = 37.77777778 + 56.25 Reorder the terms: 56.25 + 15x + x2 = 37.77777778 + 56.25 Combine like terms: 37.77777778 + 56.25 = 94.02777778 56.25 + 15x + x2 = 94.02777778 Factor a perfect square on the left side: (x + 7.5)(x + 7.5) = 94.02777778 Calculate the square root of the right side: 9.696792139 Break this problem into two subproblems by setting (x + 7.5) equal to 9.696792139 and -9.696792139.Subproblem 1
x + 7.5 = 9.696792139 Simplifying x + 7.5 = 9.696792139 Reorder the terms: 7.5 + x = 9.696792139 Solving 7.5 + x = 9.696792139 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-7.5' to each side of the equation. 7.5 + -7.5 + x = 9.696792139 + -7.5 Combine like terms: 7.5 + -7.5 = 0.0 0.0 + x = 9.696792139 + -7.5 x = 9.696792139 + -7.5 Combine like terms: 9.696792139 + -7.5 = 2.196792139 x = 2.196792139 Simplifying x = 2.196792139Subproblem 2
x + 7.5 = -9.696792139 Simplifying x + 7.5 = -9.696792139 Reorder the terms: 7.5 + x = -9.696792139 Solving 7.5 + x = -9.696792139 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-7.5' to each side of the equation. 7.5 + -7.5 + x = -9.696792139 + -7.5 Combine like terms: 7.5 + -7.5 = 0.0 0.0 + x = -9.696792139 + -7.5 x = -9.696792139 + -7.5 Combine like terms: -9.696792139 + -7.5 = -17.196792139 x = -17.196792139 Simplifying x = -17.196792139Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.196792139, -17.196792139}Solution
x = {2.196792139, -17.196792139}
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