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Simplifying (10x + 7)(4x + 5) = 180 Reorder the terms: (7 + 10x)(4x + 5) = 180 Reorder the terms: (7 + 10x)(5 + 4x) = 180 Multiply (7 + 10x) * (5 + 4x) (7(5 + 4x) + 10x * (5 + 4x)) = 180 ((5 * 7 + 4x * 7) + 10x * (5 + 4x)) = 180 ((35 + 28x) + 10x * (5 + 4x)) = 180 (35 + 28x + (5 * 10x + 4x * 10x)) = 180 (35 + 28x + (50x + 40x2)) = 180 Combine like terms: 28x + 50x = 78x (35 + 78x + 40x2) = 180 Solving 35 + 78x + 40x2 = 180 Solving for variable 'x'. Reorder the terms: 35 + -180 + 78x + 40x2 = 180 + -180 Combine like terms: 35 + -180 = -145 -145 + 78x + 40x2 = 180 + -180 Combine like terms: 180 + -180 = 0 -145 + 78x + 40x2 = 0 Begin completing the square. Divide all terms by 40 the coefficient of the squared term: Divide each side by '40'. -3.625 + 1.95x + x2 = 0.0 Move the constant term to the right: Add '3.625' to each side of the equation. -3.625 + 1.95x + 3.625 + x2 = 0.0 + 3.625 Reorder the terms: -3.625 + 3.625 + 1.95x + x2 = 0.0 + 3.625 Combine like terms: -3.625 + 3.625 = 0.000 0.000 + 1.95x + x2 = 0.0 + 3.625 1.95x + x2 = 0.0 + 3.625 Combine like terms: 0.0 + 3.625 = 3.625 1.95x + x2 = 3.625 The x term is 1.95x. Take half its coefficient (0.975). Square it (0.950625) and add it to both sides. Add '0.950625' to each side of the equation. 1.95x + 0.950625 + x2 = 3.625 + 0.950625 Reorder the terms: 0.950625 + 1.95x + x2 = 3.625 + 0.950625 Combine like terms: 3.625 + 0.950625 = 4.575625 0.950625 + 1.95x + x2 = 4.575625 Factor a perfect square on the left side: (x + 0.975)(x + 0.975) = 4.575625 Calculate the square root of the right side: 2.13907106 Break this problem into two subproblems by setting (x + 0.975) equal to 2.13907106 and -2.13907106.Subproblem 1
x + 0.975 = 2.13907106 Simplifying x + 0.975 = 2.13907106 Reorder the terms: 0.975 + x = 2.13907106 Solving 0.975 + x = 2.13907106 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.975' to each side of the equation. 0.975 + -0.975 + x = 2.13907106 + -0.975 Combine like terms: 0.975 + -0.975 = 0.000 0.000 + x = 2.13907106 + -0.975 x = 2.13907106 + -0.975 Combine like terms: 2.13907106 + -0.975 = 1.16407106 x = 1.16407106 Simplifying x = 1.16407106Subproblem 2
x + 0.975 = -2.13907106 Simplifying x + 0.975 = -2.13907106 Reorder the terms: 0.975 + x = -2.13907106 Solving 0.975 + x = -2.13907106 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.975' to each side of the equation. 0.975 + -0.975 + x = -2.13907106 + -0.975 Combine like terms: 0.975 + -0.975 = 0.000 0.000 + x = -2.13907106 + -0.975 x = -2.13907106 + -0.975 Combine like terms: -2.13907106 + -0.975 = -3.11407106 x = -3.11407106 Simplifying x = -3.11407106Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.16407106, -3.11407106}
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