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Simplifying x = 0.25((0.851(15 + -1.75x) + -2.5)(0.85(15 + -1.75x) + -2.5)) x = 0.25(((15 * 0.851 + -1.75x * 0.851) + -2.5)(0.85(15 + -1.75x) + -2.5)) x = 0.25(((12.765 + -1.48925x) + -2.5)(0.85(15 + -1.75x) + -2.5)) Reorder the terms: x = 0.25((12.765 + -2.5 + -1.48925x)(0.85(15 + -1.75x) + -2.5)) Combine like terms: 12.765 + -2.5 = 10.265 x = 0.25((10.265 + -1.48925x)(0.85(15 + -1.75x) + -2.5)) x = 0.25((10.265 + -1.48925x)((15 * 0.85 + -1.75x * 0.85) + -2.5)) x = 0.25((10.265 + -1.48925x)((12.75 + -1.4875x) + -2.5)) Reorder the terms: x = 0.25((10.265 + -1.48925x)(12.75 + -2.5 + -1.4875x)) Combine like terms: 12.75 + -2.5 = 10.25 x = 0.25((10.265 + -1.48925x)(10.25 + -1.4875x)) Multiply (10.265 + -1.48925x) * (10.25 + -1.4875x) x = 0.25((10.265(10.25 + -1.4875x) + -1.48925x * (10.25 + -1.4875x))) x = 0.25(((10.25 * 10.265 + -1.4875x * 10.265) + -1.48925x * (10.25 + -1.4875x))) x = 0.25(((105.21625 + -15.2691875x) + -1.48925x * (10.25 + -1.4875x))) x = 0.25((105.21625 + -15.2691875x + (10.25 * -1.48925x + -1.4875x * -1.48925x))) x = 0.25((105.21625 + -15.2691875x + (-15.2648125x + 2.215259375x2))) Combine like terms: -15.2691875x + -15.2648125x = -30.534x x = 0.25((105.21625 + -30.534x + 2.215259375x2)) x = (105.21625 * 0.25 + -30.534x * 0.25 + 2.215259375x2 * 0.25) x = (26.3040625 + -7.6335x + 0.5538148438x2) Solving x = 26.3040625 + -7.6335x + 0.5538148438x2 Solving for variable 'x'. Reorder the terms: -26.3040625 + x + 7.6335x + -0.5538148438x2 = 26.3040625 + -7.6335x + 0.5538148438x2 + -26.3040625 + 7.6335x + -0.5538148438x2 Combine like terms: x + 7.6335x = 8.6335x -26.3040625 + 8.6335x + -0.5538148438x2 = 26.3040625 + -7.6335x + 0.5538148438x2 + -26.3040625 + 7.6335x + -0.5538148438x2 Reorder the terms: -26.3040625 + 8.6335x + -0.5538148438x2 = 26.3040625 + -26.3040625 + -7.6335x + 7.6335x + 0.5538148438x2 + -0.5538148438x2 Combine like terms: 26.3040625 + -26.3040625 = 0.0000000 -26.3040625 + 8.6335x + -0.5538148438x2 = 0.0000000 + -7.6335x + 7.6335x + 0.5538148438x2 + -0.5538148438x2 -26.3040625 + 8.6335x + -0.5538148438x2 = -7.6335x + 7.6335x + 0.5538148438x2 + -0.5538148438x2 Combine like terms: -7.6335x + 7.6335x = 0.0000 -26.3040625 + 8.6335x + -0.5538148438x2 = 0.0000 + 0.5538148438x2 + -0.5538148438x2 -26.3040625 + 8.6335x + -0.5538148438x2 = 0.5538148438x2 + -0.5538148438x2 Combine like terms: 0.5538148438x2 + -0.5538148438x2 = 0.0000000000 -26.3040625 + 8.6335x + -0.5538148438x2 = 0.0000000000 Begin completing the square. Divide all terms by -0.5538148438 the coefficient of the squared term: Divide each side by '-0.5538148438'. 47.49613123 + -15.58914517x + x2 = 0 Move the constant term to the right: Add '-47.49613123' to each side of the equation. 47.49613123 + -15.58914517x + -47.49613123 + x2 = 0 + -47.49613123 Reorder the terms: 47.49613123 + -47.49613123 + -15.58914517x + x2 = 0 + -47.49613123 Combine like terms: 47.49613123 + -47.49613123 = 0.00000000 0.00000000 + -15.58914517x + x2 = 0 + -47.49613123 -15.58914517x + x2 = 0 + -47.49613123 Combine like terms: 0 + -47.49613123 = -47.49613123 -15.58914517x + x2 = -47.49613123 The x term is -15.58914517x. Take half its coefficient (-7.794572585). Square it (60.75536178) and add it to both sides. Add '60.75536178' to each side of the equation. -15.58914517x + 60.75536178 + x2 = -47.49613123 + 60.75536178 Reorder the terms: 60.75536178 + -15.58914517x + x2 = -47.49613123 + 60.75536178 Combine like terms: -47.49613123 + 60.75536178 = 13.25923055 60.75536178 + -15.58914517x + x2 = 13.25923055 Factor a perfect square on the left side: (x + -7.794572585)(x + -7.794572585) = 13.25923055 Calculate the square root of the right side: 3.641322637 Break this problem into two subproblems by setting (x + -7.794572585) equal to 3.641322637 and -3.641322637.Subproblem 1
x + -7.794572585 = 3.641322637 Simplifying x + -7.794572585 = 3.641322637 Reorder the terms: -7.794572585 + x = 3.641322637 Solving -7.794572585 + x = 3.641322637 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '7.794572585' to each side of the equation. -7.794572585 + 7.794572585 + x = 3.641322637 + 7.794572585 Combine like terms: -7.794572585 + 7.794572585 = 0.000000000 0.000000000 + x = 3.641322637 + 7.794572585 x = 3.641322637 + 7.794572585 Combine like terms: 3.641322637 + 7.794572585 = 11.435895222 x = 11.435895222 Simplifying x = 11.435895222Subproblem 2
x + -7.794572585 = -3.641322637 Simplifying x + -7.794572585 = -3.641322637 Reorder the terms: -7.794572585 + x = -3.641322637 Solving -7.794572585 + x = -3.641322637 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '7.794572585' to each side of the equation. -7.794572585 + 7.794572585 + x = -3.641322637 + 7.794572585 Combine like terms: -7.794572585 + 7.794572585 = 0.000000000 0.000000000 + x = -3.641322637 + 7.794572585 x = -3.641322637 + 7.794572585 Combine like terms: -3.641322637 + 7.794572585 = 4.153249948 x = 4.153249948 Simplifying x = 4.153249948Solution
The solution to the problem is based on the solutions from the subproblems. x = {11.435895222, 4.153249948}
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