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Simplifying 14x2 + 31x + -15 = 0 Reorder the terms: -15 + 31x + 14x2 = 0 Solving -15 + 31x + 14x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 14 the coefficient of the squared term: Divide each side by '14'. -1.071428571 + 2.214285714x + x2 = 0 Move the constant term to the right: Add '1.071428571' to each side of the equation. -1.071428571 + 2.214285714x + 1.071428571 + x2 = 0 + 1.071428571 Reorder the terms: -1.071428571 + 1.071428571 + 2.214285714x + x2 = 0 + 1.071428571 Combine like terms: -1.071428571 + 1.071428571 = 0.000000000 0.000000000 + 2.214285714x + x2 = 0 + 1.071428571 2.214285714x + x2 = 0 + 1.071428571 Combine like terms: 0 + 1.071428571 = 1.071428571 2.214285714x + x2 = 1.071428571 The x term is 2.214285714x. Take half its coefficient (1.107142857). Square it (1.225765306) and add it to both sides. Add '1.225765306' to each side of the equation. 2.214285714x + 1.225765306 + x2 = 1.071428571 + 1.225765306 Reorder the terms: 1.225765306 + 2.214285714x + x2 = 1.071428571 + 1.225765306 Combine like terms: 1.071428571 + 1.225765306 = 2.297193877 1.225765306 + 2.214285714x + x2 = 2.297193877 Factor a perfect square on the left side: (x + 1.107142857)(x + 1.107142857) = 2.297193877 Calculate the square root of the right side: 1.515649655 Break this problem into two subproblems by setting (x + 1.107142857) equal to 1.515649655 and -1.515649655.Subproblem 1
x + 1.107142857 = 1.515649655 Simplifying x + 1.107142857 = 1.515649655 Reorder the terms: 1.107142857 + x = 1.515649655 Solving 1.107142857 + x = 1.515649655 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.107142857' to each side of the equation. 1.107142857 + -1.107142857 + x = 1.515649655 + -1.107142857 Combine like terms: 1.107142857 + -1.107142857 = 0.000000000 0.000000000 + x = 1.515649655 + -1.107142857 x = 1.515649655 + -1.107142857 Combine like terms: 1.515649655 + -1.107142857 = 0.408506798 x = 0.408506798 Simplifying x = 0.408506798Subproblem 2
x + 1.107142857 = -1.515649655 Simplifying x + 1.107142857 = -1.515649655 Reorder the terms: 1.107142857 + x = -1.515649655 Solving 1.107142857 + x = -1.515649655 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.107142857' to each side of the equation. 1.107142857 + -1.107142857 + x = -1.515649655 + -1.107142857 Combine like terms: 1.107142857 + -1.107142857 = 0.000000000 0.000000000 + x = -1.515649655 + -1.107142857 x = -1.515649655 + -1.107142857 Combine like terms: -1.515649655 + -1.107142857 = -2.622792512 x = -2.622792512 Simplifying x = -2.622792512Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.408506798, -2.622792512}
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