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Simplifying 36(x + -5)(x + 20) + 4(x + 12)(x + -4) = 16(x + -3)(x + 6) Reorder the terms: 36(-5 + x)(x + 20) + 4(x + 12)(x + -4) = 16(x + -3)(x + 6) Reorder the terms: 36(-5 + x)(20 + x) + 4(x + 12)(x + -4) = 16(x + -3)(x + 6) Multiply (-5 + x) * (20 + x) 36(-5(20 + x) + x(20 + x)) + 4(x + 12)(x + -4) = 16(x + -3)(x + 6) 36((20 * -5 + x * -5) + x(20 + x)) + 4(x + 12)(x + -4) = 16(x + -3)(x + 6) 36((-100 + -5x) + x(20 + x)) + 4(x + 12)(x + -4) = 16(x + -3)(x + 6) 36(-100 + -5x + (20 * x + x * x)) + 4(x + 12)(x + -4) = 16(x + -3)(x + 6) 36(-100 + -5x + (20x + x2)) + 4(x + 12)(x + -4) = 16(x + -3)(x + 6) Combine like terms: -5x + 20x = 15x 36(-100 + 15x + x2) + 4(x + 12)(x + -4) = 16(x + -3)(x + 6) (-100 * 36 + 15x * 36 + x2 * 36) + 4(x + 12)(x + -4) = 16(x + -3)(x + 6) (-3600 + 540x + 36x2) + 4(x + 12)(x + -4) = 16(x + -3)(x + 6) Reorder the terms: -3600 + 540x + 36x2 + 4(12 + x)(x + -4) = 16(x + -3)(x + 6) Reorder the terms: -3600 + 540x + 36x2 + 4(12 + x)(-4 + x) = 16(x + -3)(x + 6) Multiply (12 + x) * (-4 + x) -3600 + 540x + 36x2 + 4(12(-4 + x) + x(-4 + x)) = 16(x + -3)(x + 6) -3600 + 540x + 36x2 + 4((-4 * 12 + x * 12) + x(-4 + x)) = 16(x + -3)(x + 6) -3600 + 540x + 36x2 + 4((-48 + 12x) + x(-4 + x)) = 16(x + -3)(x + 6) -3600 + 540x + 36x2 + 4(-48 + 12x + (-4 * x + x * x)) = 16(x + -3)(x + 6) -3600 + 540x + 36x2 + 4(-48 + 12x + (-4x + x2)) = 16(x + -3)(x + 6) Combine like terms: 12x + -4x = 8x -3600 + 540x + 36x2 + 4(-48 + 8x + x2) = 16(x + -3)(x + 6) -3600 + 540x + 36x2 + (-48 * 4 + 8x * 4 + x2 * 4) = 16(x + -3)(x + 6) -3600 + 540x + 36x2 + (-192 + 32x + 4x2) = 16(x + -3)(x + 6) Reorder the terms: -3600 + -192 + 540x + 32x + 36x2 + 4x2 = 16(x + -3)(x + 6) Combine like terms: -3600 + -192 = -3792 -3792 + 540x + 32x + 36x2 + 4x2 = 16(x + -3)(x + 6) Combine like terms: 540x + 32x = 572x -3792 + 572x + 36x2 + 4x2 = 16(x + -3)(x + 6) Combine like terms: 36x2 + 4x2 = 40x2 -3792 + 572x + 40x2 = 16(x + -3)(x + 6) Reorder the terms: -3792 + 572x + 40x2 = 16(-3 + x)(x + 6) Reorder the terms: -3792 + 572x + 40x2 = 16(-3 + x)(6 + x) Multiply (-3 + x) * (6 + x) -3792 + 572x + 40x2 = 16(-3(6 + x) + x(6 + x)) -3792 + 572x + 40x2 = 16((6 * -3 + x * -3) + x(6 + x)) -3792 + 572x + 40x2 = 16((-18 + -3x) + x(6 + x)) -3792 + 572x + 40x2 = 16(-18 + -3x + (6 * x + x * x)) -3792 + 572x + 40x2 = 16(-18 + -3x + (6x + x2)) Combine like terms: -3x + 6x = 3x -3792 + 572x + 40x2 = 16(-18 + 3x + x2) -3792 + 572x + 40x2 = (-18 * 16 + 3x * 16 + x2 * 16) -3792 + 572x + 40x2 = (-288 + 48x + 16x2) Solving -3792 + 572x + 40x2 = -288 + 48x + 16x2 Solving for variable 'x'. Reorder the terms: -3792 + 288 + 572x + -48x + 40x2 + -16x2 = -288 + 48x + 16x2 + 288 + -48x + -16x2 Combine like terms: -3792 + 288 = -3504 -3504 + 572x + -48x + 40x2 + -16x2 = -288 + 48x + 16x2 + 288 + -48x + -16x2 Combine like terms: 572x + -48x = 524x -3504 + 524x + 40x2 + -16x2 = -288 + 48x + 16x2 + 288 + -48x + -16x2 Combine like terms: 40x2 + -16x2 = 24x2 -3504 + 524x + 24x2 = -288 + 48x + 16x2 + 288 + -48x + -16x2 Reorder the terms: -3504 + 524x + 24x2 = -288 + 288 + 48x + -48x + 16x2 + -16x2 Combine like terms: -288 + 288 = 0 -3504 + 524x + 24x2 = 0 + 48x + -48x + 16x2 + -16x2 -3504 + 524x + 24x2 = 48x + -48x + 16x2 + -16x2 Combine like terms: 48x + -48x = 0 -3504 + 524x + 24x2 = 0 + 16x2 + -16x2 -3504 + 524x + 24x2 = 16x2 + -16x2 Combine like terms: 16x2 + -16x2 = 0 -3504 + 524x + 24x2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(-876 + 131x + 6x2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-876 + 131x + 6x2)' equal to zero and attempt to solve: Simplifying -876 + 131x + 6x2 = 0 Solving -876 + 131x + 6x2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -146 + 21.83333333x + x2 = 0 Move the constant term to the right: Add '146' to each side of the equation. -146 + 21.83333333x + 146 + x2 = 0 + 146 Reorder the terms: -146 + 146 + 21.83333333x + x2 = 0 + 146 Combine like terms: -146 + 146 = 0 0 + 21.83333333x + x2 = 0 + 146 21.83333333x + x2 = 0 + 146 Combine like terms: 0 + 146 = 146 21.83333333x + x2 = 146 The x term is 21.83333333x. Take half its coefficient (10.91666667). Square it (119.1736112) and add it to both sides. Add '119.1736112' to each side of the equation. 21.83333333x + 119.1736112 + x2 = 146 + 119.1736112 Reorder the terms: 119.1736112 + 21.83333333x + x2 = 146 + 119.1736112 Combine like terms: 146 + 119.1736112 = 265.1736112 119.1736112 + 21.83333333x + x2 = 265.1736112 Factor a perfect square on the left side: (x + 10.91666667)(x + 10.91666667) = 265.1736112 Calculate the square root of the right side: 16.284152149 Break this problem into two subproblems by setting (x + 10.91666667) equal to 16.284152149 and -16.284152149.Subproblem 1
x + 10.91666667 = 16.284152149 Simplifying x + 10.91666667 = 16.284152149 Reorder the terms: 10.91666667 + x = 16.284152149 Solving 10.91666667 + x = 16.284152149 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-10.91666667' to each side of the equation. 10.91666667 + -10.91666667 + x = 16.284152149 + -10.91666667 Combine like terms: 10.91666667 + -10.91666667 = 0.00000000 0.00000000 + x = 16.284152149 + -10.91666667 x = 16.284152149 + -10.91666667 Combine like terms: 16.284152149 + -10.91666667 = 5.367485479 x = 5.367485479 Simplifying x = 5.367485479Subproblem 2
x + 10.91666667 = -16.284152149 Simplifying x + 10.91666667 = -16.284152149 Reorder the terms: 10.91666667 + x = -16.284152149 Solving 10.91666667 + x = -16.284152149 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-10.91666667' to each side of the equation. 10.91666667 + -10.91666667 + x = -16.284152149 + -10.91666667 Combine like terms: 10.91666667 + -10.91666667 = 0.00000000 0.00000000 + x = -16.284152149 + -10.91666667 x = -16.284152149 + -10.91666667 Combine like terms: -16.284152149 + -10.91666667 = -27.200818819 x = -27.200818819 Simplifying x = -27.200818819Solution
The solution to the problem is based on the solutions from the subproblems. x = {5.367485479, -27.200818819}Solution
x = {5.367485479, -27.200818819}
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