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Simplifying 7.5(3x + 9)(8x + 7) = 0 Reorder the terms: 7.5(9 + 3x)(8x + 7) = 0 Reorder the terms: 7.5(9 + 3x)(7 + 8x) = 0 Multiply (9 + 3x) * (7 + 8x) 7.5(9(7 + 8x) + 3x * (7 + 8x)) = 0 7.5((7 * 9 + 8x * 9) + 3x * (7 + 8x)) = 0 7.5((63 + 72x) + 3x * (7 + 8x)) = 0 7.5(63 + 72x + (7 * 3x + 8x * 3x)) = 0 7.5(63 + 72x + (21x + 24x2)) = 0 Combine like terms: 72x + 21x = 93x 7.5(63 + 93x + 24x2) = 0 (63 * 7.5 + 93x * 7.5 + 24x2 * 7.5) = 0 (472.5 + 697.5x + 180x2) = 0 Solving 472.5 + 697.5x + 180x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 180 the coefficient of the squared term: Divide each side by '180'. 2.625 + 3.875x + x2 = 0.0 Move the constant term to the right: Add '-2.625' to each side of the equation. 2.625 + 3.875x + -2.625 + x2 = 0.0 + -2.625 Reorder the terms: 2.625 + -2.625 + 3.875x + x2 = 0.0 + -2.625 Combine like terms: 2.625 + -2.625 = 0.000 0.000 + 3.875x + x2 = 0.0 + -2.625 3.875x + x2 = 0.0 + -2.625 Combine like terms: 0.0 + -2.625 = -2.625 3.875x + x2 = -2.625 The x term is 3.875x. Take half its coefficient (1.9375). Square it (3.75390625) and add it to both sides. Add '3.75390625' to each side of the equation. 3.875x + 3.75390625 + x2 = -2.625 + 3.75390625 Reorder the terms: 3.75390625 + 3.875x + x2 = -2.625 + 3.75390625 Combine like terms: -2.625 + 3.75390625 = 1.12890625 3.75390625 + 3.875x + x2 = 1.12890625 Factor a perfect square on the left side: (x + 1.9375)(x + 1.9375) = 1.12890625 Calculate the square root of the right side: 1.0625 Break this problem into two subproblems by setting (x + 1.9375) equal to 1.0625 and -1.0625.Subproblem 1
x + 1.9375 = 1.0625 Simplifying x + 1.9375 = 1.0625 Reorder the terms: 1.9375 + x = 1.0625 Solving 1.9375 + x = 1.0625 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.9375' to each side of the equation. 1.9375 + -1.9375 + x = 1.0625 + -1.9375 Combine like terms: 1.9375 + -1.9375 = 0.0000 0.0000 + x = 1.0625 + -1.9375 x = 1.0625 + -1.9375 Combine like terms: 1.0625 + -1.9375 = -0.875 x = -0.875 Simplifying x = -0.875Subproblem 2
x + 1.9375 = -1.0625 Simplifying x + 1.9375 = -1.0625 Reorder the terms: 1.9375 + x = -1.0625 Solving 1.9375 + x = -1.0625 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.9375' to each side of the equation. 1.9375 + -1.9375 + x = -1.0625 + -1.9375 Combine like terms: 1.9375 + -1.9375 = 0.0000 0.0000 + x = -1.0625 + -1.9375 x = -1.0625 + -1.9375 Combine like terms: -1.0625 + -1.9375 = -3 x = -3 Simplifying x = -3Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.875, -3}
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