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Simplifying (2x + 5)(3x + -5) = 90 Reorder the terms: (5 + 2x)(3x + -5) = 90 Reorder the terms: (5 + 2x)(-5 + 3x) = 90 Multiply (5 + 2x) * (-5 + 3x) (5(-5 + 3x) + 2x * (-5 + 3x)) = 90 ((-5 * 5 + 3x * 5) + 2x * (-5 + 3x)) = 90 ((-25 + 15x) + 2x * (-5 + 3x)) = 90 (-25 + 15x + (-5 * 2x + 3x * 2x)) = 90 (-25 + 15x + (-10x + 6x2)) = 90 Combine like terms: 15x + -10x = 5x (-25 + 5x + 6x2) = 90 Solving -25 + 5x + 6x2 = 90 Solving for variable 'x'. Reorder the terms: -25 + -90 + 5x + 6x2 = 90 + -90 Combine like terms: -25 + -90 = -115 -115 + 5x + 6x2 = 90 + -90 Combine like terms: 90 + -90 = 0 -115 + 5x + 6x2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -19.16666667 + 0.8333333333x + x2 = 0 Move the constant term to the right: Add '19.16666667' to each side of the equation. -19.16666667 + 0.8333333333x + 19.16666667 + x2 = 0 + 19.16666667 Reorder the terms: -19.16666667 + 19.16666667 + 0.8333333333x + x2 = 0 + 19.16666667 Combine like terms: -19.16666667 + 19.16666667 = 0.00000000 0.00000000 + 0.8333333333x + x2 = 0 + 19.16666667 0.8333333333x + x2 = 0 + 19.16666667 Combine like terms: 0 + 19.16666667 = 19.16666667 0.8333333333x + x2 = 19.16666667 The x term is 0.8333333333x. Take half its coefficient (0.4166666667). Square it (0.1736111111) and add it to both sides. Add '0.1736111111' to each side of the equation. 0.8333333333x + 0.1736111111 + x2 = 19.16666667 + 0.1736111111 Reorder the terms: 0.1736111111 + 0.8333333333x + x2 = 19.16666667 + 0.1736111111 Combine like terms: 19.16666667 + 0.1736111111 = 19.3402777811 0.1736111111 + 0.8333333333x + x2 = 19.3402777811 Factor a perfect square on the left side: (x + 0.4166666667)(x + 0.4166666667) = 19.3402777811 Calculate the square root of the right side: 4.397758268 Break this problem into two subproblems by setting (x + 0.4166666667) equal to 4.397758268 and -4.397758268.Subproblem 1
x + 0.4166666667 = 4.397758268 Simplifying x + 0.4166666667 = 4.397758268 Reorder the terms: 0.4166666667 + x = 4.397758268 Solving 0.4166666667 + x = 4.397758268 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.4166666667' to each side of the equation. 0.4166666667 + -0.4166666667 + x = 4.397758268 + -0.4166666667 Combine like terms: 0.4166666667 + -0.4166666667 = 0.0000000000 0.0000000000 + x = 4.397758268 + -0.4166666667 x = 4.397758268 + -0.4166666667 Combine like terms: 4.397758268 + -0.4166666667 = 3.9810916013 x = 3.9810916013 Simplifying x = 3.9810916013Subproblem 2
x + 0.4166666667 = -4.397758268 Simplifying x + 0.4166666667 = -4.397758268 Reorder the terms: 0.4166666667 + x = -4.397758268 Solving 0.4166666667 + x = -4.397758268 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.4166666667' to each side of the equation. 0.4166666667 + -0.4166666667 + x = -4.397758268 + -0.4166666667 Combine like terms: 0.4166666667 + -0.4166666667 = 0.0000000000 0.0000000000 + x = -4.397758268 + -0.4166666667 x = -4.397758268 + -0.4166666667 Combine like terms: -4.397758268 + -0.4166666667 = -4.8144249347 x = -4.8144249347 Simplifying x = -4.8144249347Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.9810916013, -4.8144249347}
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