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Simplifying (8x + 26)(3x) = 180 Reorder the terms: (26 + 8x)(3x) = 180 Remove parenthesis around (3x) (26 + 8x) * 3x = 180 Reorder the terms for easier multiplication: 3x(26 + 8x) = 180 (26 * 3x + 8x * 3x) = 180 (78x + 24x2) = 180 Solving 78x + 24x2 = 180 Solving for variable 'x'. Reorder the terms: -180 + 78x + 24x2 = 180 + -180 Combine like terms: 180 + -180 = 0 -180 + 78x + 24x2 = 0 Factor out the Greatest Common Factor (GCF), '6'. 6(-30 + 13x + 4x2) = 0 Ignore the factor 6.Subproblem 1
Set the factor '(-30 + 13x + 4x2)' equal to zero and attempt to solve: Simplifying -30 + 13x + 4x2 = 0 Solving -30 + 13x + 4x2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -7.5 + 3.25x + x2 = 0 Move the constant term to the right: Add '7.5' to each side of the equation. -7.5 + 3.25x + 7.5 + x2 = 0 + 7.5 Reorder the terms: -7.5 + 7.5 + 3.25x + x2 = 0 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + 3.25x + x2 = 0 + 7.5 3.25x + x2 = 0 + 7.5 Combine like terms: 0 + 7.5 = 7.5 3.25x + x2 = 7.5 The x term is 3.25x. Take half its coefficient (1.625). Square it (2.640625) and add it to both sides. Add '2.640625' to each side of the equation. 3.25x + 2.640625 + x2 = 7.5 + 2.640625 Reorder the terms: 2.640625 + 3.25x + x2 = 7.5 + 2.640625 Combine like terms: 7.5 + 2.640625 = 10.140625 2.640625 + 3.25x + x2 = 10.140625 Factor a perfect square on the left side: (x + 1.625)(x + 1.625) = 10.140625 Calculate the square root of the right side: 3.184434801 Break this problem into two subproblems by setting (x + 1.625) equal to 3.184434801 and -3.184434801.Subproblem 1
x + 1.625 = 3.184434801 Simplifying x + 1.625 = 3.184434801 Reorder the terms: 1.625 + x = 3.184434801 Solving 1.625 + x = 3.184434801 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.625' to each side of the equation. 1.625 + -1.625 + x = 3.184434801 + -1.625 Combine like terms: 1.625 + -1.625 = 0.000 0.000 + x = 3.184434801 + -1.625 x = 3.184434801 + -1.625 Combine like terms: 3.184434801 + -1.625 = 1.559434801 x = 1.559434801 Simplifying x = 1.559434801Subproblem 2
x + 1.625 = -3.184434801 Simplifying x + 1.625 = -3.184434801 Reorder the terms: 1.625 + x = -3.184434801 Solving 1.625 + x = -3.184434801 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.625' to each side of the equation. 1.625 + -1.625 + x = -3.184434801 + -1.625 Combine like terms: 1.625 + -1.625 = 0.000 0.000 + x = -3.184434801 + -1.625 x = -3.184434801 + -1.625 Combine like terms: -3.184434801 + -1.625 = -4.809434801 x = -4.809434801 Simplifying x = -4.809434801Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.559434801, -4.809434801}Solution
x = {1.559434801, -4.809434801}
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