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Simplifying 6x2 + -9x + -6 = 1 Reorder the terms: -6 + -9x + 6x2 = 1 Solving -6 + -9x + 6x2 = 1 Solving for variable 'x'. Reorder the terms: -6 + -1 + -9x + 6x2 = 1 + -1 Combine like terms: -6 + -1 = -7 -7 + -9x + 6x2 = 1 + -1 Combine like terms: 1 + -1 = 0 -7 + -9x + 6x2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -1.166666667 + -1.5x + x2 = 0 Move the constant term to the right: Add '1.166666667' to each side of the equation. -1.166666667 + -1.5x + 1.166666667 + x2 = 0 + 1.166666667 Reorder the terms: -1.166666667 + 1.166666667 + -1.5x + x2 = 0 + 1.166666667 Combine like terms: -1.166666667 + 1.166666667 = 0.000000000 0.000000000 + -1.5x + x2 = 0 + 1.166666667 -1.5x + x2 = 0 + 1.166666667 Combine like terms: 0 + 1.166666667 = 1.166666667 -1.5x + x2 = 1.166666667 The x term is -1.5x. Take half its coefficient (-0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. -1.5x + 0.5625 + x2 = 1.166666667 + 0.5625 Reorder the terms: 0.5625 + -1.5x + x2 = 1.166666667 + 0.5625 Combine like terms: 1.166666667 + 0.5625 = 1.729166667 0.5625 + -1.5x + x2 = 1.729166667 Factor a perfect square on the left side: (x + -0.75)(x + -0.75) = 1.729166667 Calculate the square root of the right side: 1.31497782 Break this problem into two subproblems by setting (x + -0.75) equal to 1.31497782 and -1.31497782.Subproblem 1
x + -0.75 = 1.31497782 Simplifying x + -0.75 = 1.31497782 Reorder the terms: -0.75 + x = 1.31497782 Solving -0.75 + x = 1.31497782 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + x = 1.31497782 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + x = 1.31497782 + 0.75 x = 1.31497782 + 0.75 Combine like terms: 1.31497782 + 0.75 = 2.06497782 x = 2.06497782 Simplifying x = 2.06497782Subproblem 2
x + -0.75 = -1.31497782 Simplifying x + -0.75 = -1.31497782 Reorder the terms: -0.75 + x = -1.31497782 Solving -0.75 + x = -1.31497782 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + x = -1.31497782 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + x = -1.31497782 + 0.75 x = -1.31497782 + 0.75 Combine like terms: -1.31497782 + 0.75 = -0.56497782 x = -0.56497782 Simplifying x = -0.56497782Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.06497782, -0.56497782}
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