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Simplifying (4x + 3)(2x + 7) = 274 Reorder the terms: (3 + 4x)(2x + 7) = 274 Reorder the terms: (3 + 4x)(7 + 2x) = 274 Multiply (3 + 4x) * (7 + 2x) (3(7 + 2x) + 4x * (7 + 2x)) = 274 ((7 * 3 + 2x * 3) + 4x * (7 + 2x)) = 274 ((21 + 6x) + 4x * (7 + 2x)) = 274 (21 + 6x + (7 * 4x + 2x * 4x)) = 274 (21 + 6x + (28x + 8x2)) = 274 Combine like terms: 6x + 28x = 34x (21 + 34x + 8x2) = 274 Solving 21 + 34x + 8x2 = 274 Solving for variable 'x'. Reorder the terms: 21 + -274 + 34x + 8x2 = 274 + -274 Combine like terms: 21 + -274 = -253 -253 + 34x + 8x2 = 274 + -274 Combine like terms: 274 + -274 = 0 -253 + 34x + 8x2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -31.625 + 4.25x + x2 = 0 Move the constant term to the right: Add '31.625' to each side of the equation. -31.625 + 4.25x + 31.625 + x2 = 0 + 31.625 Reorder the terms: -31.625 + 31.625 + 4.25x + x2 = 0 + 31.625 Combine like terms: -31.625 + 31.625 = 0.000 0.000 + 4.25x + x2 = 0 + 31.625 4.25x + x2 = 0 + 31.625 Combine like terms: 0 + 31.625 = 31.625 4.25x + x2 = 31.625 The x term is 4.25x. Take half its coefficient (2.125). Square it (4.515625) and add it to both sides. Add '4.515625' to each side of the equation. 4.25x + 4.515625 + x2 = 31.625 + 4.515625 Reorder the terms: 4.515625 + 4.25x + x2 = 31.625 + 4.515625 Combine like terms: 31.625 + 4.515625 = 36.140625 4.515625 + 4.25x + x2 = 36.140625 Factor a perfect square on the left side: (x + 2.125)(x + 2.125) = 36.140625 Calculate the square root of the right side: 6.011707328 Break this problem into two subproblems by setting (x + 2.125) equal to 6.011707328 and -6.011707328.Subproblem 1
x + 2.125 = 6.011707328 Simplifying x + 2.125 = 6.011707328 Reorder the terms: 2.125 + x = 6.011707328 Solving 2.125 + x = 6.011707328 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.125' to each side of the equation. 2.125 + -2.125 + x = 6.011707328 + -2.125 Combine like terms: 2.125 + -2.125 = 0.000 0.000 + x = 6.011707328 + -2.125 x = 6.011707328 + -2.125 Combine like terms: 6.011707328 + -2.125 = 3.886707328 x = 3.886707328 Simplifying x = 3.886707328Subproblem 2
x + 2.125 = -6.011707328 Simplifying x + 2.125 = -6.011707328 Reorder the terms: 2.125 + x = -6.011707328 Solving 2.125 + x = -6.011707328 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.125' to each side of the equation. 2.125 + -2.125 + x = -6.011707328 + -2.125 Combine like terms: 2.125 + -2.125 = 0.000 0.000 + x = -6.011707328 + -2.125 x = -6.011707328 + -2.125 Combine like terms: -6.011707328 + -2.125 = -8.136707328 x = -8.136707328 Simplifying x = -8.136707328Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.886707328, -8.136707328}
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