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Simplifying (2x + 7)(2x + 10) + 2x(2x) = 342 Reorder the terms: (7 + 2x)(2x + 10) + 2x(2x) = 342 Reorder the terms: (7 + 2x)(10 + 2x) + 2x(2x) = 342 Multiply (7 + 2x) * (10 + 2x) (7(10 + 2x) + 2x * (10 + 2x)) + 2x(2x) = 342 ((10 * 7 + 2x * 7) + 2x * (10 + 2x)) + 2x(2x) = 342 ((70 + 14x) + 2x * (10 + 2x)) + 2x(2x) = 342 (70 + 14x + (10 * 2x + 2x * 2x)) + 2x(2x) = 342 (70 + 14x + (20x + 4x2)) + 2x(2x) = 342 Combine like terms: 14x + 20x = 34x (70 + 34x + 4x2) + 2x(2x) = 342 Remove parenthesis around (2x) 70 + 34x + 4x2 + 2x * 2x = 342 Reorder the terms for easier multiplication: 70 + 34x + 4x2 + 2 * 2x * x = 342 Multiply 2 * 2 70 + 34x + 4x2 + 4x * x = 342 Multiply x * x 70 + 34x + 4x2 + 4x2 = 342 Combine like terms: 4x2 + 4x2 = 8x2 70 + 34x + 8x2 = 342 Solving 70 + 34x + 8x2 = 342 Solving for variable 'x'. Reorder the terms: 70 + -342 + 34x + 8x2 = 342 + -342 Combine like terms: 70 + -342 = -272 -272 + 34x + 8x2 = 342 + -342 Combine like terms: 342 + -342 = 0 -272 + 34x + 8x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-136 + 17x + 4x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-136 + 17x + 4x2)' equal to zero and attempt to solve: Simplifying -136 + 17x + 4x2 = 0 Solving -136 + 17x + 4x2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -34 + 4.25x + x2 = 0 Move the constant term to the right: Add '34' to each side of the equation. -34 + 4.25x + 34 + x2 = 0 + 34 Reorder the terms: -34 + 34 + 4.25x + x2 = 0 + 34 Combine like terms: -34 + 34 = 0 0 + 4.25x + x2 = 0 + 34 4.25x + x2 = 0 + 34 Combine like terms: 0 + 34 = 34 4.25x + x2 = 34 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 = 34 + 4.515625 Reorder the terms: 4.515625 + 4.25x + x2 = 34 + 4.515625 Combine like terms: 34 + 4.515625 = 38.515625 4.515625 + 4.25x + x2 = 38.515625 Factor a perfect square on the left side: (x + 2.125)(x + 2.125) = 38.515625 Calculate the square root of the right side: 6.206095794 Break this problem into two subproblems by setting (x + 2.125) equal to 6.206095794 and -6.206095794.Subproblem 1
x + 2.125 = 6.206095794 Simplifying x + 2.125 = 6.206095794 Reorder the terms: 2.125 + x = 6.206095794 Solving 2.125 + x = 6.206095794 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 = 6.206095794 + -2.125 Combine like terms: 2.125 + -2.125 = 0.000 0.000 + x = 6.206095794 + -2.125 x = 6.206095794 + -2.125 Combine like terms: 6.206095794 + -2.125 = 4.081095794 x = 4.081095794 Simplifying x = 4.081095794Subproblem 2
x + 2.125 = -6.206095794 Simplifying x + 2.125 = -6.206095794 Reorder the terms: 2.125 + x = -6.206095794 Solving 2.125 + x = -6.206095794 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 = -6.206095794 + -2.125 Combine like terms: 2.125 + -2.125 = 0.000 0.000 + x = -6.206095794 + -2.125 x = -6.206095794 + -2.125 Combine like terms: -6.206095794 + -2.125 = -8.331095794 x = -8.331095794 Simplifying x = -8.331095794Solution
The solution to the problem is based on the solutions from the subproblems. x = {4.081095794, -8.331095794}Solution
x = {4.081095794, -8.331095794}
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