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Simplifying 0 = 3x2 + -5x + -13 Reorder the terms: 0 = -13 + -5x + 3x2 Solving 0 = -13 + -5x + 3x2 Solving for variable 'x'. Combine like terms: 0 + 13 = 13 13 + 5x + -3x2 = -13 + -5x + 3x2 + 13 + 5x + -3x2 Reorder the terms: 13 + 5x + -3x2 = -13 + 13 + -5x + 5x + 3x2 + -3x2 Combine like terms: -13 + 13 = 0 13 + 5x + -3x2 = 0 + -5x + 5x + 3x2 + -3x2 13 + 5x + -3x2 = -5x + 5x + 3x2 + -3x2 Combine like terms: -5x + 5x = 0 13 + 5x + -3x2 = 0 + 3x2 + -3x2 13 + 5x + -3x2 = 3x2 + -3x2 Combine like terms: 3x2 + -3x2 = 0 13 + 5x + -3x2 = 0 Begin completing the square. Divide all terms by -3 the coefficient of the squared term: Divide each side by '-3'. -4.333333333 + -1.666666667x + x2 = 0 Move the constant term to the right: Add '4.333333333' to each side of the equation. -4.333333333 + -1.666666667x + 4.333333333 + x2 = 0 + 4.333333333 Reorder the terms: -4.333333333 + 4.333333333 + -1.666666667x + x2 = 0 + 4.333333333 Combine like terms: -4.333333333 + 4.333333333 = 0.000000000 0.000000000 + -1.666666667x + x2 = 0 + 4.333333333 -1.666666667x + x2 = 0 + 4.333333333 Combine like terms: 0 + 4.333333333 = 4.333333333 -1.666666667x + x2 = 4.333333333 The x term is -1.666666667x. Take half its coefficient (-0.8333333335). Square it (0.6944444447) and add it to both sides. Add '0.6944444447' to each side of the equation. -1.666666667x + 0.6944444447 + x2 = 4.333333333 + 0.6944444447 Reorder the terms: 0.6944444447 + -1.666666667x + x2 = 4.333333333 + 0.6944444447 Combine like terms: 4.333333333 + 0.6944444447 = 5.0277777777 0.6944444447 + -1.666666667x + x2 = 5.0277777777 Factor a perfect square on the left side: (x + -0.8333333335)(x + -0.8333333335) = 5.0277777777 Calculate the square root of the right side: 2.242270674 Break this problem into two subproblems by setting (x + -0.8333333335) equal to 2.242270674 and -2.242270674.Subproblem 1
x + -0.8333333335 = 2.242270674 Simplifying x + -0.8333333335 = 2.242270674 Reorder the terms: -0.8333333335 + x = 2.242270674 Solving -0.8333333335 + x = 2.242270674 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.8333333335' to each side of the equation. -0.8333333335 + 0.8333333335 + x = 2.242270674 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + x = 2.242270674 + 0.8333333335 x = 2.242270674 + 0.8333333335 Combine like terms: 2.242270674 + 0.8333333335 = 3.0756040075 x = 3.0756040075 Simplifying x = 3.0756040075Subproblem 2
x + -0.8333333335 = -2.242270674 Simplifying x + -0.8333333335 = -2.242270674 Reorder the terms: -0.8333333335 + x = -2.242270674 Solving -0.8333333335 + x = -2.242270674 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.8333333335' to each side of the equation. -0.8333333335 + 0.8333333335 + x = -2.242270674 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + x = -2.242270674 + 0.8333333335 x = -2.242270674 + 0.8333333335 Combine like terms: -2.242270674 + 0.8333333335 = -1.4089373405 x = -1.4089373405 Simplifying x = -1.4089373405Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.0756040075, -1.4089373405}
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