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Simplifying 5(x + 5)(2x + 1) = 110 Reorder the terms: 5(5 + x)(2x + 1) = 110 Reorder the terms: 5(5 + x)(1 + 2x) = 110 Multiply (5 + x) * (1 + 2x) 5(5(1 + 2x) + x(1 + 2x)) = 110 5((1 * 5 + 2x * 5) + x(1 + 2x)) = 110 5((5 + 10x) + x(1 + 2x)) = 110 5(5 + 10x + (1 * x + 2x * x)) = 110 5(5 + 10x + (1x + 2x2)) = 110 Combine like terms: 10x + 1x = 11x 5(5 + 11x + 2x2) = 110 (5 * 5 + 11x * 5 + 2x2 * 5) = 110 (25 + 55x + 10x2) = 110 Solving 25 + 55x + 10x2 = 110 Solving for variable 'x'. Reorder the terms: 25 + -110 + 55x + 10x2 = 110 + -110 Combine like terms: 25 + -110 = -85 -85 + 55x + 10x2 = 110 + -110 Combine like terms: 110 + -110 = 0 -85 + 55x + 10x2 = 0 Factor out the Greatest Common Factor (GCF), '5'. 5(-17 + 11x + 2x2) = 0 Ignore the factor 5.Subproblem 1
Set the factor '(-17 + 11x + 2x2)' equal to zero and attempt to solve: Simplifying -17 + 11x + 2x2 = 0 Solving -17 + 11x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -8.5 + 5.5x + x2 = 0 Move the constant term to the right: Add '8.5' to each side of the equation. -8.5 + 5.5x + 8.5 + x2 = 0 + 8.5 Reorder the terms: -8.5 + 8.5 + 5.5x + x2 = 0 + 8.5 Combine like terms: -8.5 + 8.5 = 0.0 0.0 + 5.5x + x2 = 0 + 8.5 5.5x + x2 = 0 + 8.5 Combine like terms: 0 + 8.5 = 8.5 5.5x + x2 = 8.5 The x term is 5.5x. Take half its coefficient (2.75). Square it (7.5625) and add it to both sides. Add '7.5625' to each side of the equation. 5.5x + 7.5625 + x2 = 8.5 + 7.5625 Reorder the terms: 7.5625 + 5.5x + x2 = 8.5 + 7.5625 Combine like terms: 8.5 + 7.5625 = 16.0625 7.5625 + 5.5x + x2 = 16.0625 Factor a perfect square on the left side: (x + 2.75)(x + 2.75) = 16.0625 Calculate the square root of the right side: 4.007804885 Break this problem into two subproblems by setting (x + 2.75) equal to 4.007804885 and -4.007804885.Subproblem 1
x + 2.75 = 4.007804885 Simplifying x + 2.75 = 4.007804885 Reorder the terms: 2.75 + x = 4.007804885 Solving 2.75 + x = 4.007804885 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.75' to each side of the equation. 2.75 + -2.75 + x = 4.007804885 + -2.75 Combine like terms: 2.75 + -2.75 = 0.00 0.00 + x = 4.007804885 + -2.75 x = 4.007804885 + -2.75 Combine like terms: 4.007804885 + -2.75 = 1.257804885 x = 1.257804885 Simplifying x = 1.257804885Subproblem 2
x + 2.75 = -4.007804885 Simplifying x + 2.75 = -4.007804885 Reorder the terms: 2.75 + x = -4.007804885 Solving 2.75 + x = -4.007804885 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.75' to each side of the equation. 2.75 + -2.75 + x = -4.007804885 + -2.75 Combine like terms: 2.75 + -2.75 = 0.00 0.00 + x = -4.007804885 + -2.75 x = -4.007804885 + -2.75 Combine like terms: -4.007804885 + -2.75 = -6.757804885 x = -6.757804885 Simplifying x = -6.757804885Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.257804885, -6.757804885}Solution
x = {1.257804885, -6.757804885}
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