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Simplifying (2x + 1)(x + 4) * 13 = 2x + -10x Reorder the terms: (1 + 2x)(x + 4) * 13 = 2x + -10x Reorder the terms: (1 + 2x)(4 + x) * 13 = 2x + -10x Reorder the terms for easier multiplication: 13(1 + 2x)(4 + x) = 2x + -10x Multiply (1 + 2x) * (4 + x) 13(1(4 + x) + 2x * (4 + x)) = 2x + -10x 13((4 * 1 + x * 1) + 2x * (4 + x)) = 2x + -10x 13((4 + 1x) + 2x * (4 + x)) = 2x + -10x 13(4 + 1x + (4 * 2x + x * 2x)) = 2x + -10x 13(4 + 1x + (8x + 2x2)) = 2x + -10x Combine like terms: 1x + 8x = 9x 13(4 + 9x + 2x2) = 2x + -10x (4 * 13 + 9x * 13 + 2x2 * 13) = 2x + -10x (52 + 117x + 26x2) = 2x + -10x Combine like terms: 2x + -10x = -8x 52 + 117x + 26x2 = -8x Solving 52 + 117x + 26x2 = -8x Solving for variable 'x'. Reorder the terms: 52 + 117x + 8x + 26x2 = -8x + 8x Combine like terms: 117x + 8x = 125x 52 + 125x + 26x2 = -8x + 8x Combine like terms: -8x + 8x = 0 52 + 125x + 26x2 = 0 Begin completing the square. Divide all terms by 26 the coefficient of the squared term: Divide each side by '26'. 2 + 4.807692308x + x2 = 0 Move the constant term to the right: Add '-2' to each side of the equation. 2 + 4.807692308x + -2 + x2 = 0 + -2 Reorder the terms: 2 + -2 + 4.807692308x + x2 = 0 + -2 Combine like terms: 2 + -2 = 0 0 + 4.807692308x + x2 = 0 + -2 4.807692308x + x2 = 0 + -2 Combine like terms: 0 + -2 = -2 4.807692308x + x2 = -2 The x term is 4.807692308x. Take half its coefficient (2.403846154). Square it (5.778476332) and add it to both sides. Add '5.778476332' to each side of the equation. 4.807692308x + 5.778476332 + x2 = -2 + 5.778476332 Reorder the terms: 5.778476332 + 4.807692308x + x2 = -2 + 5.778476332 Combine like terms: -2 + 5.778476332 = 3.778476332 5.778476332 + 4.807692308x + x2 = 3.778476332 Factor a perfect square on the left side: (x + 2.403846154)(x + 2.403846154) = 3.778476332 Calculate the square root of the right side: 1.943830325 Break this problem into two subproblems by setting (x + 2.403846154) equal to 1.943830325 and -1.943830325.Subproblem 1
x + 2.403846154 = 1.943830325 Simplifying x + 2.403846154 = 1.943830325 Reorder the terms: 2.403846154 + x = 1.943830325 Solving 2.403846154 + x = 1.943830325 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.403846154' to each side of the equation. 2.403846154 + -2.403846154 + x = 1.943830325 + -2.403846154 Combine like terms: 2.403846154 + -2.403846154 = 0.000000000 0.000000000 + x = 1.943830325 + -2.403846154 x = 1.943830325 + -2.403846154 Combine like terms: 1.943830325 + -2.403846154 = -0.460015829 x = -0.460015829 Simplifying x = -0.460015829Subproblem 2
x + 2.403846154 = -1.943830325 Simplifying x + 2.403846154 = -1.943830325 Reorder the terms: 2.403846154 + x = -1.943830325 Solving 2.403846154 + x = -1.943830325 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.403846154' to each side of the equation. 2.403846154 + -2.403846154 + x = -1.943830325 + -2.403846154 Combine like terms: 2.403846154 + -2.403846154 = 0.000000000 0.000000000 + x = -1.943830325 + -2.403846154 x = -1.943830325 + -2.403846154 Combine like terms: -1.943830325 + -2.403846154 = -4.347676479 x = -4.347676479 Simplifying x = -4.347676479Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.460015829, -4.347676479}
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