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Simplifying (2x + 21)(3x + -4) = 90 Reorder the terms: (21 + 2x)(3x + -4) = 90 Reorder the terms: (21 + 2x)(-4 + 3x) = 90 Multiply (21 + 2x) * (-4 + 3x) (21(-4 + 3x) + 2x * (-4 + 3x)) = 90 ((-4 * 21 + 3x * 21) + 2x * (-4 + 3x)) = 90 ((-84 + 63x) + 2x * (-4 + 3x)) = 90 (-84 + 63x + (-4 * 2x + 3x * 2x)) = 90 (-84 + 63x + (-8x + 6x2)) = 90 Combine like terms: 63x + -8x = 55x (-84 + 55x + 6x2) = 90 Solving -84 + 55x + 6x2 = 90 Solving for variable 'x'. Reorder the terms: -84 + -90 + 55x + 6x2 = 90 + -90 Combine like terms: -84 + -90 = -174 -174 + 55x + 6x2 = 90 + -90 Combine like terms: 90 + -90 = 0 -174 + 55x + 6x2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -29 + 9.166666667x + x2 = 0 Move the constant term to the right: Add '29' to each side of the equation. -29 + 9.166666667x + 29 + x2 = 0 + 29 Reorder the terms: -29 + 29 + 9.166666667x + x2 = 0 + 29 Combine like terms: -29 + 29 = 0 0 + 9.166666667x + x2 = 0 + 29 9.166666667x + x2 = 0 + 29 Combine like terms: 0 + 29 = 29 9.166666667x + x2 = 29 The x term is 9.166666667x. Take half its coefficient (4.583333334). Square it (21.00694445) and add it to both sides. Add '21.00694445' to each side of the equation. 9.166666667x + 21.00694445 + x2 = 29 + 21.00694445 Reorder the terms: 21.00694445 + 9.166666667x + x2 = 29 + 21.00694445 Combine like terms: 29 + 21.00694445 = 50.00694445 21.00694445 + 9.166666667x + x2 = 50.00694445 Factor a perfect square on the left side: (x + 4.583333334)(x + 4.583333334) = 50.00694445 Calculate the square root of the right side: 7.071558842 Break this problem into two subproblems by setting (x + 4.583333334) equal to 7.071558842 and -7.071558842.Subproblem 1
x + 4.583333334 = 7.071558842 Simplifying x + 4.583333334 = 7.071558842 Reorder the terms: 4.583333334 + x = 7.071558842 Solving 4.583333334 + x = 7.071558842 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-4.583333334' to each side of the equation. 4.583333334 + -4.583333334 + x = 7.071558842 + -4.583333334 Combine like terms: 4.583333334 + -4.583333334 = 0.000000000 0.000000000 + x = 7.071558842 + -4.583333334 x = 7.071558842 + -4.583333334 Combine like terms: 7.071558842 + -4.583333334 = 2.488225508 x = 2.488225508 Simplifying x = 2.488225508Subproblem 2
x + 4.583333334 = -7.071558842 Simplifying x + 4.583333334 = -7.071558842 Reorder the terms: 4.583333334 + x = -7.071558842 Solving 4.583333334 + x = -7.071558842 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-4.583333334' to each side of the equation. 4.583333334 + -4.583333334 + x = -7.071558842 + -4.583333334 Combine like terms: 4.583333334 + -4.583333334 = 0.000000000 0.000000000 + x = -7.071558842 + -4.583333334 x = -7.071558842 + -4.583333334 Combine like terms: -7.071558842 + -4.583333334 = -11.654892176 x = -11.654892176 Simplifying x = -11.654892176Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.488225508, -11.654892176}
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