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Simplifying (12x + 1)(7x + 8) = 180 Reorder the terms: (1 + 12x)(7x + 8) = 180 Reorder the terms: (1 + 12x)(8 + 7x) = 180 Multiply (1 + 12x) * (8 + 7x) (1(8 + 7x) + 12x * (8 + 7x)) = 180 ((8 * 1 + 7x * 1) + 12x * (8 + 7x)) = 180 ((8 + 7x) + 12x * (8 + 7x)) = 180 (8 + 7x + (8 * 12x + 7x * 12x)) = 180 (8 + 7x + (96x + 84x2)) = 180 Combine like terms: 7x + 96x = 103x (8 + 103x + 84x2) = 180 Solving 8 + 103x + 84x2 = 180 Solving for variable 'x'. Reorder the terms: 8 + -180 + 103x + 84x2 = 180 + -180 Combine like terms: 8 + -180 = -172 -172 + 103x + 84x2 = 180 + -180 Combine like terms: 180 + -180 = 0 -172 + 103x + 84x2 = 0 Begin completing the square. Divide all terms by 84 the coefficient of the squared term: Divide each side by '84'. -2.047619048 + 1.226190476x + x2 = 0 Move the constant term to the right: Add '2.047619048' to each side of the equation. -2.047619048 + 1.226190476x + 2.047619048 + x2 = 0 + 2.047619048 Reorder the terms: -2.047619048 + 2.047619048 + 1.226190476x + x2 = 0 + 2.047619048 Combine like terms: -2.047619048 + 2.047619048 = 0.000000000 0.000000000 + 1.226190476x + x2 = 0 + 2.047619048 1.226190476x + x2 = 0 + 2.047619048 Combine like terms: 0 + 2.047619048 = 2.047619048 1.226190476x + x2 = 2.047619048 The x term is 1.226190476x. Take half its coefficient (0.613095238). Square it (0.3758857709) and add it to both sides. Add '0.3758857709' to each side of the equation. 1.226190476x + 0.3758857709 + x2 = 2.047619048 + 0.3758857709 Reorder the terms: 0.3758857709 + 1.226190476x + x2 = 2.047619048 + 0.3758857709 Combine like terms: 2.047619048 + 0.3758857709 = 2.4235048189 0.3758857709 + 1.226190476x + x2 = 2.4235048189 Factor a perfect square on the left side: (x + 0.613095238)(x + 0.613095238) = 2.4235048189 Calculate the square root of the right side: 1.556761002 Break this problem into two subproblems by setting (x + 0.613095238) equal to 1.556761002 and -1.556761002.Subproblem 1
x + 0.613095238 = 1.556761002 Simplifying x + 0.613095238 = 1.556761002 Reorder the terms: 0.613095238 + x = 1.556761002 Solving 0.613095238 + x = 1.556761002 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.613095238' to each side of the equation. 0.613095238 + -0.613095238 + x = 1.556761002 + -0.613095238 Combine like terms: 0.613095238 + -0.613095238 = 0.000000000 0.000000000 + x = 1.556761002 + -0.613095238 x = 1.556761002 + -0.613095238 Combine like terms: 1.556761002 + -0.613095238 = 0.943665764 x = 0.943665764 Simplifying x = 0.943665764Subproblem 2
x + 0.613095238 = -1.556761002 Simplifying x + 0.613095238 = -1.556761002 Reorder the terms: 0.613095238 + x = -1.556761002 Solving 0.613095238 + x = -1.556761002 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.613095238' to each side of the equation. 0.613095238 + -0.613095238 + x = -1.556761002 + -0.613095238 Combine like terms: 0.613095238 + -0.613095238 = 0.000000000 0.000000000 + x = -1.556761002 + -0.613095238 x = -1.556761002 + -0.613095238 Combine like terms: -1.556761002 + -0.613095238 = -2.16985624 x = -2.16985624 Simplifying x = -2.16985624Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.943665764, -2.16985624}
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