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Simplifying (3x + 8)(6x + -22) = 180 Reorder the terms: (8 + 3x)(6x + -22) = 180 Reorder the terms: (8 + 3x)(-22 + 6x) = 180 Multiply (8 + 3x) * (-22 + 6x) (8(-22 + 6x) + 3x * (-22 + 6x)) = 180 ((-22 * 8 + 6x * 8) + 3x * (-22 + 6x)) = 180 ((-176 + 48x) + 3x * (-22 + 6x)) = 180 (-176 + 48x + (-22 * 3x + 6x * 3x)) = 180 (-176 + 48x + (-66x + 18x2)) = 180 Combine like terms: 48x + -66x = -18x (-176 + -18x + 18x2) = 180 Solving -176 + -18x + 18x2 = 180 Solving for variable 'x'. Reorder the terms: -176 + -180 + -18x + 18x2 = 180 + -180 Combine like terms: -176 + -180 = -356 -356 + -18x + 18x2 = 180 + -180 Combine like terms: 180 + -180 = 0 -356 + -18x + 18x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-178 + -9x + 9x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-178 + -9x + 9x2)' equal to zero and attempt to solve: Simplifying -178 + -9x + 9x2 = 0 Solving -178 + -9x + 9x2 = 0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -19.77777778 + -1x + x2 = 0 Move the constant term to the right: Add '19.77777778' to each side of the equation. -19.77777778 + -1x + 19.77777778 + x2 = 0 + 19.77777778 Reorder the terms: -19.77777778 + 19.77777778 + -1x + x2 = 0 + 19.77777778 Combine like terms: -19.77777778 + 19.77777778 = 0.00000000 0.00000000 + -1x + x2 = 0 + 19.77777778 -1x + x2 = 0 + 19.77777778 Combine like terms: 0 + 19.77777778 = 19.77777778 -1x + x2 = 19.77777778 The x term is -1x. Take half its coefficient (-0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -1x + 0.25 + x2 = 19.77777778 + 0.25 Reorder the terms: 0.25 + -1x + x2 = 19.77777778 + 0.25 Combine like terms: 19.77777778 + 0.25 = 20.02777778 0.25 + -1x + x2 = 20.02777778 Factor a perfect square on the left side: (x + -0.5)(x + -0.5) = 20.02777778 Calculate the square root of the right side: 4.475240528 Break this problem into two subproblems by setting (x + -0.5) equal to 4.475240528 and -4.475240528.Subproblem 1
x + -0.5 = 4.475240528 Simplifying x + -0.5 = 4.475240528 Reorder the terms: -0.5 + x = 4.475240528 Solving -0.5 + x = 4.475240528 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + x = 4.475240528 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = 4.475240528 + 0.5 x = 4.475240528 + 0.5 Combine like terms: 4.475240528 + 0.5 = 4.975240528 x = 4.975240528 Simplifying x = 4.975240528Subproblem 2
x + -0.5 = -4.475240528 Simplifying x + -0.5 = -4.475240528 Reorder the terms: -0.5 + x = -4.475240528 Solving -0.5 + x = -4.475240528 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + x = -4.475240528 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = -4.475240528 + 0.5 x = -4.475240528 + 0.5 Combine like terms: -4.475240528 + 0.5 = -3.975240528 x = -3.975240528 Simplifying x = -3.975240528Solution
The solution to the problem is based on the solutions from the subproblems. x = {4.975240528, -3.975240528}Solution
x = {4.975240528, -3.975240528}
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