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Simplifying (6x + -7)(9x + -8) = 180 Reorder the terms: (-7 + 6x)(9x + -8) = 180 Reorder the terms: (-7 + 6x)(-8 + 9x) = 180 Multiply (-7 + 6x) * (-8 + 9x) (-7(-8 + 9x) + 6x * (-8 + 9x)) = 180 ((-8 * -7 + 9x * -7) + 6x * (-8 + 9x)) = 180 ((56 + -63x) + 6x * (-8 + 9x)) = 180 (56 + -63x + (-8 * 6x + 9x * 6x)) = 180 (56 + -63x + (-48x + 54x2)) = 180 Combine like terms: -63x + -48x = -111x (56 + -111x + 54x2) = 180 Solving 56 + -111x + 54x2 = 180 Solving for variable 'x'. Reorder the terms: 56 + -180 + -111x + 54x2 = 180 + -180 Combine like terms: 56 + -180 = -124 -124 + -111x + 54x2 = 180 + -180 Combine like terms: 180 + -180 = 0 -124 + -111x + 54x2 = 0 Begin completing the square. Divide all terms by 54 the coefficient of the squared term: Divide each side by '54'. -2.296296296 + -2.055555556x + x2 = 0 Move the constant term to the right: Add '2.296296296' to each side of the equation. -2.296296296 + -2.055555556x + 2.296296296 + x2 = 0 + 2.296296296 Reorder the terms: -2.296296296 + 2.296296296 + -2.055555556x + x2 = 0 + 2.296296296 Combine like terms: -2.296296296 + 2.296296296 = 0.000000000 0.000000000 + -2.055555556x + x2 = 0 + 2.296296296 -2.055555556x + x2 = 0 + 2.296296296 Combine like terms: 0 + 2.296296296 = 2.296296296 -2.055555556x + x2 = 2.296296296 The x term is -2.055555556x. Take half its coefficient (-1.027777778). Square it (1.056327161) and add it to both sides. Add '1.056327161' to each side of the equation. -2.055555556x + 1.056327161 + x2 = 2.296296296 + 1.056327161 Reorder the terms: 1.056327161 + -2.055555556x + x2 = 2.296296296 + 1.056327161 Combine like terms: 2.296296296 + 1.056327161 = 3.352623457 1.056327161 + -2.055555556x + x2 = 3.352623457 Factor a perfect square on the left side: (x + -1.027777778)(x + -1.027777778) = 3.352623457 Calculate the square root of the right side: 1.831017055 Break this problem into two subproblems by setting (x + -1.027777778) equal to 1.831017055 and -1.831017055.Subproblem 1
x + -1.027777778 = 1.831017055 Simplifying x + -1.027777778 = 1.831017055 Reorder the terms: -1.027777778 + x = 1.831017055 Solving -1.027777778 + x = 1.831017055 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.027777778' to each side of the equation. -1.027777778 + 1.027777778 + x = 1.831017055 + 1.027777778 Combine like terms: -1.027777778 + 1.027777778 = 0.000000000 0.000000000 + x = 1.831017055 + 1.027777778 x = 1.831017055 + 1.027777778 Combine like terms: 1.831017055 + 1.027777778 = 2.858794833 x = 2.858794833 Simplifying x = 2.858794833Subproblem 2
x + -1.027777778 = -1.831017055 Simplifying x + -1.027777778 = -1.831017055 Reorder the terms: -1.027777778 + x = -1.831017055 Solving -1.027777778 + x = -1.831017055 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.027777778' to each side of the equation. -1.027777778 + 1.027777778 + x = -1.831017055 + 1.027777778 Combine like terms: -1.027777778 + 1.027777778 = 0.000000000 0.000000000 + x = -1.831017055 + 1.027777778 x = -1.831017055 + 1.027777778 Combine like terms: -1.831017055 + 1.027777778 = -0.803239277 x = -0.803239277 Simplifying x = -0.803239277Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.858794833, -0.803239277}
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