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Simplifying (12x + 1)(7x + 8) = 90 Reorder the terms: (1 + 12x)(7x + 8) = 90 Reorder the terms: (1 + 12x)(8 + 7x) = 90 Multiply (1 + 12x) * (8 + 7x) (1(8 + 7x) + 12x * (8 + 7x)) = 90 ((8 * 1 + 7x * 1) + 12x * (8 + 7x)) = 90 ((8 + 7x) + 12x * (8 + 7x)) = 90 (8 + 7x + (8 * 12x + 7x * 12x)) = 90 (8 + 7x + (96x + 84x2)) = 90 Combine like terms: 7x + 96x = 103x (8 + 103x + 84x2) = 90 Solving 8 + 103x + 84x2 = 90 Solving for variable 'x'. Reorder the terms: 8 + -90 + 103x + 84x2 = 90 + -90 Combine like terms: 8 + -90 = -82 -82 + 103x + 84x2 = 90 + -90 Combine like terms: 90 + -90 = 0 -82 + 103x + 84x2 = 0 Begin completing the square. Divide all terms by 84 the coefficient of the squared term: Divide each side by '84'. -0.9761904762 + 1.226190476x + x2 = 0 Move the constant term to the right: Add '0.9761904762' to each side of the equation. -0.9761904762 + 1.226190476x + 0.9761904762 + x2 = 0 + 0.9761904762 Reorder the terms: -0.9761904762 + 0.9761904762 + 1.226190476x + x2 = 0 + 0.9761904762 Combine like terms: -0.9761904762 + 0.9761904762 = 0.0000000000 0.0000000000 + 1.226190476x + x2 = 0 + 0.9761904762 1.226190476x + x2 = 0 + 0.9761904762 Combine like terms: 0 + 0.9761904762 = 0.9761904762 1.226190476x + x2 = 0.9761904762 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 = 0.9761904762 + 0.3758857709 Reorder the terms: 0.3758857709 + 1.226190476x + x2 = 0.9761904762 + 0.3758857709 Combine like terms: 0.9761904762 + 0.3758857709 = 1.3520762471 0.3758857709 + 1.226190476x + x2 = 1.3520762471 Factor a perfect square on the left side: (x + 0.613095238)(x + 0.613095238) = 1.3520762471 Calculate the square root of the right side: 1.162788135 Break this problem into two subproblems by setting (x + 0.613095238) equal to 1.162788135 and -1.162788135.Subproblem 1
x + 0.613095238 = 1.162788135 Simplifying x + 0.613095238 = 1.162788135 Reorder the terms: 0.613095238 + x = 1.162788135 Solving 0.613095238 + x = 1.162788135 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 = 1.162788135 + -0.613095238 Combine like terms: 0.613095238 + -0.613095238 = 0.000000000 0.000000000 + x = 1.162788135 + -0.613095238 x = 1.162788135 + -0.613095238 Combine like terms: 1.162788135 + -0.613095238 = 0.549692897 x = 0.549692897 Simplifying x = 0.549692897Subproblem 2
x + 0.613095238 = -1.162788135 Simplifying x + 0.613095238 = -1.162788135 Reorder the terms: 0.613095238 + x = -1.162788135 Solving 0.613095238 + x = -1.162788135 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 = -1.162788135 + -0.613095238 Combine like terms: 0.613095238 + -0.613095238 = 0.000000000 0.000000000 + x = -1.162788135 + -0.613095238 x = -1.162788135 + -0.613095238 Combine like terms: -1.162788135 + -0.613095238 = -1.775883373 x = -1.775883373 Simplifying x = -1.775883373Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.549692897, -1.775883373}
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