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Simplifying (3x2 + -6x + 23) + (8x2 + 8x + -88) = 0 Reorder the terms: (23 + -6x + 3x2) + (8x2 + 8x + -88) = 0 Remove parenthesis around (23 + -6x + 3x2) 23 + -6x + 3x2 + (8x2 + 8x + -88) = 0 Reorder the terms: 23 + -6x + 3x2 + (-88 + 8x + 8x2) = 0 Remove parenthesis around (-88 + 8x + 8x2) 23 + -6x + 3x2 + -88 + 8x + 8x2 = 0 Reorder the terms: 23 + -88 + -6x + 8x + 3x2 + 8x2 = 0 Combine like terms: 23 + -88 = -65 -65 + -6x + 8x + 3x2 + 8x2 = 0 Combine like terms: -6x + 8x = 2x -65 + 2x + 3x2 + 8x2 = 0 Combine like terms: 3x2 + 8x2 = 11x2 -65 + 2x + 11x2 = 0 Solving -65 + 2x + 11x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 11 the coefficient of the squared term: Divide each side by '11'. -5.909090909 + 0.1818181818x + x2 = 0 Move the constant term to the right: Add '5.909090909' to each side of the equation. -5.909090909 + 0.1818181818x + 5.909090909 + x2 = 0 + 5.909090909 Reorder the terms: -5.909090909 + 5.909090909 + 0.1818181818x + x2 = 0 + 5.909090909 Combine like terms: -5.909090909 + 5.909090909 = 0.000000000 0.000000000 + 0.1818181818x + x2 = 0 + 5.909090909 0.1818181818x + x2 = 0 + 5.909090909 Combine like terms: 0 + 5.909090909 = 5.909090909 0.1818181818x + x2 = 5.909090909 The x term is 0.1818181818x. Take half its coefficient (0.0909090909). Square it (0.008264462808) and add it to both sides. Add '0.008264462808' to each side of the equation. 0.1818181818x + 0.008264462808 + x2 = 5.909090909 + 0.008264462808 Reorder the terms: 0.008264462808 + 0.1818181818x + x2 = 5.909090909 + 0.008264462808 Combine like terms: 5.909090909 + 0.008264462808 = 5.917355371808 0.008264462808 + 0.1818181818x + x2 = 5.917355371808 Factor a perfect square on the left side: (x + 0.0909090909)(x + 0.0909090909) = 5.917355371808 Calculate the square root of the right side: 2.432561484 Break this problem into two subproblems by setting (x + 0.0909090909) equal to 2.432561484 and -2.432561484.Subproblem 1
x + 0.0909090909 = 2.432561484 Simplifying x + 0.0909090909 = 2.432561484 Reorder the terms: 0.0909090909 + x = 2.432561484 Solving 0.0909090909 + x = 2.432561484 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.0909090909' to each side of the equation. 0.0909090909 + -0.0909090909 + x = 2.432561484 + -0.0909090909 Combine like terms: 0.0909090909 + -0.0909090909 = 0.0000000000 0.0000000000 + x = 2.432561484 + -0.0909090909 x = 2.432561484 + -0.0909090909 Combine like terms: 2.432561484 + -0.0909090909 = 2.3416523931 x = 2.3416523931 Simplifying x = 2.3416523931Subproblem 2
x + 0.0909090909 = -2.432561484 Simplifying x + 0.0909090909 = -2.432561484 Reorder the terms: 0.0909090909 + x = -2.432561484 Solving 0.0909090909 + x = -2.432561484 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.0909090909' to each side of the equation. 0.0909090909 + -0.0909090909 + x = -2.432561484 + -0.0909090909 Combine like terms: 0.0909090909 + -0.0909090909 = 0.0000000000 0.0000000000 + x = -2.432561484 + -0.0909090909 x = -2.432561484 + -0.0909090909 Combine like terms: -2.432561484 + -0.0909090909 = -2.5234705749 x = -2.5234705749 Simplifying x = -2.5234705749Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.3416523931, -2.5234705749}
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