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Simplifying (8x + 3)(5x + 8) = 180 Reorder the terms: (3 + 8x)(5x + 8) = 180 Reorder the terms: (3 + 8x)(8 + 5x) = 180 Multiply (3 + 8x) * (8 + 5x) (3(8 + 5x) + 8x * (8 + 5x)) = 180 ((8 * 3 + 5x * 3) + 8x * (8 + 5x)) = 180 ((24 + 15x) + 8x * (8 + 5x)) = 180 (24 + 15x + (8 * 8x + 5x * 8x)) = 180 (24 + 15x + (64x + 40x2)) = 180 Combine like terms: 15x + 64x = 79x (24 + 79x + 40x2) = 180 Solving 24 + 79x + 40x2 = 180 Solving for variable 'x'. Reorder the terms: 24 + -180 + 79x + 40x2 = 180 + -180 Combine like terms: 24 + -180 = -156 -156 + 79x + 40x2 = 180 + -180 Combine like terms: 180 + -180 = 0 -156 + 79x + 40x2 = 0 Begin completing the square. Divide all terms by 40 the coefficient of the squared term: Divide each side by '40'. -3.9 + 1.975x + x2 = 0.0 Move the constant term to the right: Add '3.9' to each side of the equation. -3.9 + 1.975x + 3.9 + x2 = 0.0 + 3.9 Reorder the terms: -3.9 + 3.9 + 1.975x + x2 = 0.0 + 3.9 Combine like terms: -3.9 + 3.9 = 0.0 0.0 + 1.975x + x2 = 0.0 + 3.9 1.975x + x2 = 0.0 + 3.9 Combine like terms: 0.0 + 3.9 = 3.9 1.975x + x2 = 3.9 The x term is 1.975x. Take half its coefficient (0.9875). Square it (0.97515625) and add it to both sides. Add '0.97515625' to each side of the equation. 1.975x + 0.97515625 + x2 = 3.9 + 0.97515625 Reorder the terms: 0.97515625 + 1.975x + x2 = 3.9 + 0.97515625 Combine like terms: 3.9 + 0.97515625 = 4.87515625 0.97515625 + 1.975x + x2 = 4.87515625 Factor a perfect square on the left side: (x + 0.9875)(x + 0.9875) = 4.87515625 Calculate the square root of the right side: 2.2079756 Break this problem into two subproblems by setting (x + 0.9875) equal to 2.2079756 and -2.2079756.Subproblem 1
x + 0.9875 = 2.2079756 Simplifying x + 0.9875 = 2.2079756 Reorder the terms: 0.9875 + x = 2.2079756 Solving 0.9875 + x = 2.2079756 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.9875' to each side of the equation. 0.9875 + -0.9875 + x = 2.2079756 + -0.9875 Combine like terms: 0.9875 + -0.9875 = 0.0000 0.0000 + x = 2.2079756 + -0.9875 x = 2.2079756 + -0.9875 Combine like terms: 2.2079756 + -0.9875 = 1.2204756 x = 1.2204756 Simplifying x = 1.2204756Subproblem 2
x + 0.9875 = -2.2079756 Simplifying x + 0.9875 = -2.2079756 Reorder the terms: 0.9875 + x = -2.2079756 Solving 0.9875 + x = -2.2079756 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.9875' to each side of the equation. 0.9875 + -0.9875 + x = -2.2079756 + -0.9875 Combine like terms: 0.9875 + -0.9875 = 0.0000 0.0000 + x = -2.2079756 + -0.9875 x = -2.2079756 + -0.9875 Combine like terms: -2.2079756 + -0.9875 = -3.1954756 x = -3.1954756 Simplifying x = -3.1954756Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.2204756, -3.1954756}
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