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Simplifying 14x2 + 53x + 17 = 0 Reorder the terms: 17 + 53x + 14x2 = 0 Solving 17 + 53x + 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'. 1.214285714 + 3.785714286x + x2 = 0 Move the constant term to the right: Add '-1.214285714' to each side of the equation. 1.214285714 + 3.785714286x + -1.214285714 + x2 = 0 + -1.214285714 Reorder the terms: 1.214285714 + -1.214285714 + 3.785714286x + x2 = 0 + -1.214285714 Combine like terms: 1.214285714 + -1.214285714 = 0.000000000 0.000000000 + 3.785714286x + x2 = 0 + -1.214285714 3.785714286x + x2 = 0 + -1.214285714 Combine like terms: 0 + -1.214285714 = -1.214285714 3.785714286x + x2 = -1.214285714 The x term is 3.785714286x. Take half its coefficient (1.892857143). Square it (3.582908164) and add it to both sides. Add '3.582908164' to each side of the equation. 3.785714286x + 3.582908164 + x2 = -1.214285714 + 3.582908164 Reorder the terms: 3.582908164 + 3.785714286x + x2 = -1.214285714 + 3.582908164 Combine like terms: -1.214285714 + 3.582908164 = 2.36862245 3.582908164 + 3.785714286x + x2 = 2.36862245 Factor a perfect square on the left side: (x + 1.892857143)(x + 1.892857143) = 2.36862245 Calculate the square root of the right side: 1.539032959 Break this problem into two subproblems by setting (x + 1.892857143) equal to 1.539032959 and -1.539032959.Subproblem 1
x + 1.892857143 = 1.539032959 Simplifying x + 1.892857143 = 1.539032959 Reorder the terms: 1.892857143 + x = 1.539032959 Solving 1.892857143 + x = 1.539032959 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.892857143' to each side of the equation. 1.892857143 + -1.892857143 + x = 1.539032959 + -1.892857143 Combine like terms: 1.892857143 + -1.892857143 = 0.000000000 0.000000000 + x = 1.539032959 + -1.892857143 x = 1.539032959 + -1.892857143 Combine like terms: 1.539032959 + -1.892857143 = -0.353824184 x = -0.353824184 Simplifying x = -0.353824184Subproblem 2
x + 1.892857143 = -1.539032959 Simplifying x + 1.892857143 = -1.539032959 Reorder the terms: 1.892857143 + x = -1.539032959 Solving 1.892857143 + x = -1.539032959 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.892857143' to each side of the equation. 1.892857143 + -1.892857143 + x = -1.539032959 + -1.892857143 Combine like terms: 1.892857143 + -1.892857143 = 0.000000000 0.000000000 + x = -1.539032959 + -1.892857143 x = -1.539032959 + -1.892857143 Combine like terms: -1.539032959 + -1.892857143 = -3.431890102 x = -3.431890102 Simplifying x = -3.431890102Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.353824184, -3.431890102}
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