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Simplifying 13x2 + 32x + -85 = 0 Reorder the terms: -85 + 32x + 13x2 = 0 Solving -85 + 32x + 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'. -6.538461538 + 2.461538462x + x2 = 0 Move the constant term to the right: Add '6.538461538' to each side of the equation. -6.538461538 + 2.461538462x + 6.538461538 + x2 = 0 + 6.538461538 Reorder the terms: -6.538461538 + 6.538461538 + 2.461538462x + x2 = 0 + 6.538461538 Combine like terms: -6.538461538 + 6.538461538 = 0.000000000 0.000000000 + 2.461538462x + x2 = 0 + 6.538461538 2.461538462x + x2 = 0 + 6.538461538 Combine like terms: 0 + 6.538461538 = 6.538461538 2.461538462x + x2 = 6.538461538 The x term is 2.461538462x. Take half its coefficient (1.230769231). Square it (1.514792900) and add it to both sides. Add '1.514792900' to each side of the equation. 2.461538462x + 1.514792900 + x2 = 6.538461538 + 1.514792900 Reorder the terms: 1.514792900 + 2.461538462x + x2 = 6.538461538 + 1.514792900 Combine like terms: 6.538461538 + 1.514792900 = 8.053254438 1.514792900 + 2.461538462x + x2 = 8.053254438 Factor a perfect square on the left side: (x + 1.230769231)(x + 1.230769231) = 8.053254438 Calculate the square root of the right side: 2.837825653 Break this problem into two subproblems by setting (x + 1.230769231) equal to 2.837825653 and -2.837825653.Subproblem 1
x + 1.230769231 = 2.837825653 Simplifying x + 1.230769231 = 2.837825653 Reorder the terms: 1.230769231 + x = 2.837825653 Solving 1.230769231 + x = 2.837825653 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.230769231' to each side of the equation. 1.230769231 + -1.230769231 + x = 2.837825653 + -1.230769231 Combine like terms: 1.230769231 + -1.230769231 = 0.000000000 0.000000000 + x = 2.837825653 + -1.230769231 x = 2.837825653 + -1.230769231 Combine like terms: 2.837825653 + -1.230769231 = 1.607056422 x = 1.607056422 Simplifying x = 1.607056422Subproblem 2
x + 1.230769231 = -2.837825653 Simplifying x + 1.230769231 = -2.837825653 Reorder the terms: 1.230769231 + x = -2.837825653 Solving 1.230769231 + x = -2.837825653 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.230769231' to each side of the equation. 1.230769231 + -1.230769231 + x = -2.837825653 + -1.230769231 Combine like terms: 1.230769231 + -1.230769231 = 0.000000000 0.000000000 + x = -2.837825653 + -1.230769231 x = -2.837825653 + -1.230769231 Combine like terms: -2.837825653 + -1.230769231 = -4.068594884 x = -4.068594884 Simplifying x = -4.068594884Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.607056422, -4.068594884}
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