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Simplifying (3x + 11) * 90(4x + -5) = 180 Reorder the terms: (11 + 3x) * 90(4x + -5) = 180 Reorder the terms: (11 + 3x) * 90(-5 + 4x) = 180 Reorder the terms for easier multiplication: 90(11 + 3x)(-5 + 4x) = 180 Multiply (11 + 3x) * (-5 + 4x) 90(11(-5 + 4x) + 3x * (-5 + 4x)) = 180 90((-5 * 11 + 4x * 11) + 3x * (-5 + 4x)) = 180 90((-55 + 44x) + 3x * (-5 + 4x)) = 180 90(-55 + 44x + (-5 * 3x + 4x * 3x)) = 180 90(-55 + 44x + (-15x + 12x2)) = 180 Combine like terms: 44x + -15x = 29x 90(-55 + 29x + 12x2) = 180 (-55 * 90 + 29x * 90 + 12x2 * 90) = 180 (-4950 + 2610x + 1080x2) = 180 Solving -4950 + 2610x + 1080x2 = 180 Solving for variable 'x'. Reorder the terms: -4950 + -180 + 2610x + 1080x2 = 180 + -180 Combine like terms: -4950 + -180 = -5130 -5130 + 2610x + 1080x2 = 180 + -180 Combine like terms: 180 + -180 = 0 -5130 + 2610x + 1080x2 = 0 Factor out the Greatest Common Factor (GCF), '90'. 90(-57 + 29x + 12x2) = 0 Ignore the factor 90.Subproblem 1
Set the factor '(-57 + 29x + 12x2)' equal to zero and attempt to solve: Simplifying -57 + 29x + 12x2 = 0 Solving -57 + 29x + 12x2 = 0 Begin completing the square. Divide all terms by 12 the coefficient of the squared term: Divide each side by '12'. -4.75 + 2.416666667x + x2 = 0 Move the constant term to the right: Add '4.75' to each side of the equation. -4.75 + 2.416666667x + 4.75 + x2 = 0 + 4.75 Reorder the terms: -4.75 + 4.75 + 2.416666667x + x2 = 0 + 4.75 Combine like terms: -4.75 + 4.75 = 0.00 0.00 + 2.416666667x + x2 = 0 + 4.75 2.416666667x + x2 = 0 + 4.75 Combine like terms: 0 + 4.75 = 4.75 2.416666667x + x2 = 4.75 The x term is 2.416666667x. Take half its coefficient (1.208333334). Square it (1.460069446) and add it to both sides. Add '1.460069446' to each side of the equation. 2.416666667x + 1.460069446 + x2 = 4.75 + 1.460069446 Reorder the terms: 1.460069446 + 2.416666667x + x2 = 4.75 + 1.460069446 Combine like terms: 4.75 + 1.460069446 = 6.210069446 1.460069446 + 2.416666667x + x2 = 6.210069446 Factor a perfect square on the left side: (x + 1.208333334)(x + 1.208333334) = 6.210069446 Calculate the square root of the right side: 2.492001093 Break this problem into two subproblems by setting (x + 1.208333334) equal to 2.492001093 and -2.492001093.Subproblem 1
x + 1.208333334 = 2.492001093 Simplifying x + 1.208333334 = 2.492001093 Reorder the terms: 1.208333334 + x = 2.492001093 Solving 1.208333334 + x = 2.492001093 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.208333334' to each side of the equation. 1.208333334 + -1.208333334 + x = 2.492001093 + -1.208333334 Combine like terms: 1.208333334 + -1.208333334 = 0.000000000 0.000000000 + x = 2.492001093 + -1.208333334 x = 2.492001093 + -1.208333334 Combine like terms: 2.492001093 + -1.208333334 = 1.283667759 x = 1.283667759 Simplifying x = 1.283667759Subproblem 2
x + 1.208333334 = -2.492001093 Simplifying x + 1.208333334 = -2.492001093 Reorder the terms: 1.208333334 + x = -2.492001093 Solving 1.208333334 + x = -2.492001093 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.208333334' to each side of the equation. 1.208333334 + -1.208333334 + x = -2.492001093 + -1.208333334 Combine like terms: 1.208333334 + -1.208333334 = 0.000000000 0.000000000 + x = -2.492001093 + -1.208333334 x = -2.492001093 + -1.208333334 Combine like terms: -2.492001093 + -1.208333334 = -3.700334427 x = -3.700334427 Simplifying x = -3.700334427Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.283667759, -3.700334427}Solution
x = {1.283667759, -3.700334427}
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