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Simplifying 6x2 + 23x + 18 = 0 Reorder the terms: 18 + 23x + 6x2 = 0 Solving 18 + 23x + 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 + 3.833333333x + x2 = 0 Move the constant term to the right: Add '-3' to each side of the equation. 3 + 3.833333333x + -3 + x2 = 0 + -3 Reorder the terms: 3 + -3 + 3.833333333x + x2 = 0 + -3 Combine like terms: 3 + -3 = 0 0 + 3.833333333x + x2 = 0 + -3 3.833333333x + x2 = 0 + -3 Combine like terms: 0 + -3 = -3 3.833333333x + x2 = -3 The x term is 3.833333333x. Take half its coefficient (1.916666667). Square it (3.673611112) and add it to both sides. Add '3.673611112' to each side of the equation. 3.833333333x + 3.673611112 + x2 = -3 + 3.673611112 Reorder the terms: 3.673611112 + 3.833333333x + x2 = -3 + 3.673611112 Combine like terms: -3 + 3.673611112 = 0.673611112 3.673611112 + 3.833333333x + x2 = 0.673611112 Factor a perfect square on the left side: (x + 1.916666667)(x + 1.916666667) = 0.673611112 Calculate the square root of the right side: 0.820738151 Break this problem into two subproblems by setting (x + 1.916666667) equal to 0.820738151 and -0.820738151.Subproblem 1
x + 1.916666667 = 0.820738151 Simplifying x + 1.916666667 = 0.820738151 Reorder the terms: 1.916666667 + x = 0.820738151 Solving 1.916666667 + x = 0.820738151 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.916666667' to each side of the equation. 1.916666667 + -1.916666667 + x = 0.820738151 + -1.916666667 Combine like terms: 1.916666667 + -1.916666667 = 0.000000000 0.000000000 + x = 0.820738151 + -1.916666667 x = 0.820738151 + -1.916666667 Combine like terms: 0.820738151 + -1.916666667 = -1.095928516 x = -1.095928516 Simplifying x = -1.095928516Subproblem 2
x + 1.916666667 = -0.820738151 Simplifying x + 1.916666667 = -0.820738151 Reorder the terms: 1.916666667 + x = -0.820738151 Solving 1.916666667 + x = -0.820738151 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.916666667' to each side of the equation. 1.916666667 + -1.916666667 + x = -0.820738151 + -1.916666667 Combine like terms: 1.916666667 + -1.916666667 = 0.000000000 0.000000000 + x = -0.820738151 + -1.916666667 x = -0.820738151 + -1.916666667 Combine like terms: -0.820738151 + -1.916666667 = -2.737404818 x = -2.737404818 Simplifying x = -2.737404818Solution
The solution to the problem is based on the solutions from the subproblems. x = {-1.095928516, -2.737404818}
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