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Simplifying 2x2 + 9x + -82 = 0 Reorder the terms: -82 + 9x + 2x2 = 0 Solving -82 + 9x + 2x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -41 + 4.5x + x2 = 0 Move the constant term to the right: Add '41' to each side of the equation. -41 + 4.5x + 41 + x2 = 0 + 41 Reorder the terms: -41 + 41 + 4.5x + x2 = 0 + 41 Combine like terms: -41 + 41 = 0 0 + 4.5x + x2 = 0 + 41 4.5x + x2 = 0 + 41 Combine like terms: 0 + 41 = 41 4.5x + x2 = 41 The x term is 4.5x. Take half its coefficient (2.25). Square it (5.0625) and add it to both sides. Add '5.0625' to each side of the equation. 4.5x + 5.0625 + x2 = 41 + 5.0625 Reorder the terms: 5.0625 + 4.5x + x2 = 41 + 5.0625 Combine like terms: 41 + 5.0625 = 46.0625 5.0625 + 4.5x + x2 = 46.0625 Factor a perfect square on the left side: (x + 2.25)(x + 2.25) = 46.0625 Calculate the square root of the right side: 6.78693598 Break this problem into two subproblems by setting (x + 2.25) equal to 6.78693598 and -6.78693598.Subproblem 1
x + 2.25 = 6.78693598 Simplifying x + 2.25 = 6.78693598 Reorder the terms: 2.25 + x = 6.78693598 Solving 2.25 + x = 6.78693598 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.25' to each side of the equation. 2.25 + -2.25 + x = 6.78693598 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + x = 6.78693598 + -2.25 x = 6.78693598 + -2.25 Combine like terms: 6.78693598 + -2.25 = 4.53693598 x = 4.53693598 Simplifying x = 4.53693598Subproblem 2
x + 2.25 = -6.78693598 Simplifying x + 2.25 = -6.78693598 Reorder the terms: 2.25 + x = -6.78693598 Solving 2.25 + x = -6.78693598 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.25' to each side of the equation. 2.25 + -2.25 + x = -6.78693598 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + x = -6.78693598 + -2.25 x = -6.78693598 + -2.25 Combine like terms: -6.78693598 + -2.25 = -9.03693598 x = -9.03693598 Simplifying x = -9.03693598Solution
The solution to the problem is based on the solutions from the subproblems. x = {4.53693598, -9.03693598}
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