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Simplifying 6x2 + -9x + -23 = 0 Reorder the terms: -23 + -9x + 6x2 = 0 Solving -23 + -9x + 6x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -3.833333333 + -1.5x + x2 = 0 Move the constant term to the right: Add '3.833333333' to each side of the equation. -3.833333333 + -1.5x + 3.833333333 + x2 = 0 + 3.833333333 Reorder the terms: -3.833333333 + 3.833333333 + -1.5x + x2 = 0 + 3.833333333 Combine like terms: -3.833333333 + 3.833333333 = 0.000000000 0.000000000 + -1.5x + x2 = 0 + 3.833333333 -1.5x + x2 = 0 + 3.833333333 Combine like terms: 0 + 3.833333333 = 3.833333333 -1.5x + x2 = 3.833333333 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 = 3.833333333 + 0.5625 Reorder the terms: 0.5625 + -1.5x + x2 = 3.833333333 + 0.5625 Combine like terms: 3.833333333 + 0.5625 = 4.395833333 0.5625 + -1.5x + x2 = 4.395833333 Factor a perfect square on the left side: (x + -0.75)(x + -0.75) = 4.395833333 Calculate the square root of the right side: 2.096624271 Break this problem into two subproblems by setting (x + -0.75) equal to 2.096624271 and -2.096624271.Subproblem 1
x + -0.75 = 2.096624271 Simplifying x + -0.75 = 2.096624271 Reorder the terms: -0.75 + x = 2.096624271 Solving -0.75 + x = 2.096624271 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 = 2.096624271 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + x = 2.096624271 + 0.75 x = 2.096624271 + 0.75 Combine like terms: 2.096624271 + 0.75 = 2.846624271 x = 2.846624271 Simplifying x = 2.846624271Subproblem 2
x + -0.75 = -2.096624271 Simplifying x + -0.75 = -2.096624271 Reorder the terms: -0.75 + x = -2.096624271 Solving -0.75 + x = -2.096624271 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 = -2.096624271 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + x = -2.096624271 + 0.75 x = -2.096624271 + 0.75 Combine like terms: -2.096624271 + 0.75 = -1.346624271 x = -1.346624271 Simplifying x = -1.346624271Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.846624271, -1.346624271}
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