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Simplifying 40x2 + 9x + -106 = 0 Reorder the terms: -106 + 9x + 40x2 = 0 Solving -106 + 9x + 40x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 40 the coefficient of the squared term: Divide each side by '40'. -2.65 + 0.225x + x2 = 0 Move the constant term to the right: Add '2.65' to each side of the equation. -2.65 + 0.225x + 2.65 + x2 = 0 + 2.65 Reorder the terms: -2.65 + 2.65 + 0.225x + x2 = 0 + 2.65 Combine like terms: -2.65 + 2.65 = 0.00 0.00 + 0.225x + x2 = 0 + 2.65 0.225x + x2 = 0 + 2.65 Combine like terms: 0 + 2.65 = 2.65 0.225x + x2 = 2.65 The x term is 0.225x. Take half its coefficient (0.1125). Square it (0.01265625) and add it to both sides. Add '0.01265625' to each side of the equation. 0.225x + 0.01265625 + x2 = 2.65 + 0.01265625 Reorder the terms: 0.01265625 + 0.225x + x2 = 2.65 + 0.01265625 Combine like terms: 2.65 + 0.01265625 = 2.66265625 0.01265625 + 0.225x + x2 = 2.66265625 Factor a perfect square on the left side: (x + 0.1125)(x + 0.1125) = 2.66265625 Calculate the square root of the right side: 1.631764766 Break this problem into two subproblems by setting (x + 0.1125) equal to 1.631764766 and -1.631764766.Subproblem 1
x + 0.1125 = 1.631764766 Simplifying x + 0.1125 = 1.631764766 Reorder the terms: 0.1125 + x = 1.631764766 Solving 0.1125 + x = 1.631764766 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1125' to each side of the equation. 0.1125 + -0.1125 + x = 1.631764766 + -0.1125 Combine like terms: 0.1125 + -0.1125 = 0.0000 0.0000 + x = 1.631764766 + -0.1125 x = 1.631764766 + -0.1125 Combine like terms: 1.631764766 + -0.1125 = 1.519264766 x = 1.519264766 Simplifying x = 1.519264766Subproblem 2
x + 0.1125 = -1.631764766 Simplifying x + 0.1125 = -1.631764766 Reorder the terms: 0.1125 + x = -1.631764766 Solving 0.1125 + x = -1.631764766 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1125' to each side of the equation. 0.1125 + -0.1125 + x = -1.631764766 + -0.1125 Combine like terms: 0.1125 + -0.1125 = 0.0000 0.0000 + x = -1.631764766 + -0.1125 x = -1.631764766 + -0.1125 Combine like terms: -1.631764766 + -0.1125 = -1.744264766 x = -1.744264766 Simplifying x = -1.744264766Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.519264766, -1.744264766}
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