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Simplifying 14x2 + -13x + -52 = 0 Reorder the terms: -52 + -13x + 14x2 = 0 Solving -52 + -13x + 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'. -3.714285714 + -0.9285714286x + x2 = 0 Move the constant term to the right: Add '3.714285714' to each side of the equation. -3.714285714 + -0.9285714286x + 3.714285714 + x2 = 0 + 3.714285714 Reorder the terms: -3.714285714 + 3.714285714 + -0.9285714286x + x2 = 0 + 3.714285714 Combine like terms: -3.714285714 + 3.714285714 = 0.000000000 0.000000000 + -0.9285714286x + x2 = 0 + 3.714285714 -0.9285714286x + x2 = 0 + 3.714285714 Combine like terms: 0 + 3.714285714 = 3.714285714 -0.9285714286x + x2 = 3.714285714 The x term is -0.9285714286x. Take half its coefficient (-0.4642857143). Square it (0.2155612245) and add it to both sides. Add '0.2155612245' to each side of the equation. -0.9285714286x + 0.2155612245 + x2 = 3.714285714 + 0.2155612245 Reorder the terms: 0.2155612245 + -0.9285714286x + x2 = 3.714285714 + 0.2155612245 Combine like terms: 3.714285714 + 0.2155612245 = 3.9298469385 0.2155612245 + -0.9285714286x + x2 = 3.9298469385 Factor a perfect square on the left side: (x + -0.4642857143)(x + -0.4642857143) = 3.9298469385 Calculate the square root of the right side: 1.982384155 Break this problem into two subproblems by setting (x + -0.4642857143) equal to 1.982384155 and -1.982384155.Subproblem 1
x + -0.4642857143 = 1.982384155 Simplifying x + -0.4642857143 = 1.982384155 Reorder the terms: -0.4642857143 + x = 1.982384155 Solving -0.4642857143 + x = 1.982384155 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.4642857143' to each side of the equation. -0.4642857143 + 0.4642857143 + x = 1.982384155 + 0.4642857143 Combine like terms: -0.4642857143 + 0.4642857143 = 0.0000000000 0.0000000000 + x = 1.982384155 + 0.4642857143 x = 1.982384155 + 0.4642857143 Combine like terms: 1.982384155 + 0.4642857143 = 2.4466698693 x = 2.4466698693 Simplifying x = 2.4466698693Subproblem 2
x + -0.4642857143 = -1.982384155 Simplifying x + -0.4642857143 = -1.982384155 Reorder the terms: -0.4642857143 + x = -1.982384155 Solving -0.4642857143 + x = -1.982384155 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.4642857143' to each side of the equation. -0.4642857143 + 0.4642857143 + x = -1.982384155 + 0.4642857143 Combine like terms: -0.4642857143 + 0.4642857143 = 0.0000000000 0.0000000000 + x = -1.982384155 + 0.4642857143 x = -1.982384155 + 0.4642857143 Combine like terms: -1.982384155 + 0.4642857143 = -1.5180984407 x = -1.5180984407 Simplifying x = -1.5180984407Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.4466698693, -1.5180984407}
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