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Simplifying 13x2 + 2x + -68 = 0 Reorder the terms: -68 + 2x + 13x2 = 0 Solving -68 + 2x + 13x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 13 the coefficient of the squared term: Divide each side by '13'. -5.230769231 + 0.1538461538x + x2 = 0 Move the constant term to the right: Add '5.230769231' to each side of the equation. -5.230769231 + 0.1538461538x + 5.230769231 + x2 = 0 + 5.230769231 Reorder the terms: -5.230769231 + 5.230769231 + 0.1538461538x + x2 = 0 + 5.230769231 Combine like terms: -5.230769231 + 5.230769231 = 0.000000000 0.000000000 + 0.1538461538x + x2 = 0 + 5.230769231 0.1538461538x + x2 = 0 + 5.230769231 Combine like terms: 0 + 5.230769231 = 5.230769231 0.1538461538x + x2 = 5.230769231 The x term is 0.1538461538x. Take half its coefficient (0.0769230769). Square it (0.005917159760) and add it to both sides. Add '0.005917159760' to each side of the equation. 0.1538461538x + 0.005917159760 + x2 = 5.230769231 + 0.005917159760 Reorder the terms: 0.005917159760 + 0.1538461538x + x2 = 5.230769231 + 0.005917159760 Combine like terms: 5.230769231 + 0.005917159760 = 5.23668639076 0.005917159760 + 0.1538461538x + x2 = 5.23668639076 Factor a perfect square on the left side: (x + 0.0769230769)(x + 0.0769230769) = 5.23668639076 Calculate the square root of the right side: 2.288380736 Break this problem into two subproblems by setting (x + 0.0769230769) equal to 2.288380736 and -2.288380736.Subproblem 1
x + 0.0769230769 = 2.288380736 Simplifying x + 0.0769230769 = 2.288380736 Reorder the terms: 0.0769230769 + x = 2.288380736 Solving 0.0769230769 + x = 2.288380736 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.0769230769' to each side of the equation. 0.0769230769 + -0.0769230769 + x = 2.288380736 + -0.0769230769 Combine like terms: 0.0769230769 + -0.0769230769 = 0.0000000000 0.0000000000 + x = 2.288380736 + -0.0769230769 x = 2.288380736 + -0.0769230769 Combine like terms: 2.288380736 + -0.0769230769 = 2.2114576591 x = 2.2114576591 Simplifying x = 2.2114576591Subproblem 2
x + 0.0769230769 = -2.288380736 Simplifying x + 0.0769230769 = -2.288380736 Reorder the terms: 0.0769230769 + x = -2.288380736 Solving 0.0769230769 + x = -2.288380736 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.0769230769' to each side of the equation. 0.0769230769 + -0.0769230769 + x = -2.288380736 + -0.0769230769 Combine like terms: 0.0769230769 + -0.0769230769 = 0.0000000000 0.0000000000 + x = -2.288380736 + -0.0769230769 x = -2.288380736 + -0.0769230769 Combine like terms: -2.288380736 + -0.0769230769 = -2.3653038129 x = -2.3653038129 Simplifying x = -2.3653038129Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.2114576591, -2.3653038129}
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