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Simplifying 6x2 + 12x + -82 = 0 Reorder the terms: -82 + 12x + 6x2 = 0 Solving -82 + 12x + 6x2 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '2'. 2(-41 + 6x + 3x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-41 + 6x + 3x2)' equal to zero and attempt to solve: Simplifying -41 + 6x + 3x2 = 0 Solving -41 + 6x + 3x2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -13.66666667 + 2x + x2 = 0 Move the constant term to the right: Add '13.66666667' to each side of the equation. -13.66666667 + 2x + 13.66666667 + x2 = 0 + 13.66666667 Reorder the terms: -13.66666667 + 13.66666667 + 2x + x2 = 0 + 13.66666667 Combine like terms: -13.66666667 + 13.66666667 = 0.00000000 0.00000000 + 2x + x2 = 0 + 13.66666667 2x + x2 = 0 + 13.66666667 Combine like terms: 0 + 13.66666667 = 13.66666667 2x + x2 = 13.66666667 The x term is 2x. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2x + 1 + x2 = 13.66666667 + 1 Reorder the terms: 1 + 2x + x2 = 13.66666667 + 1 Combine like terms: 13.66666667 + 1 = 14.66666667 1 + 2x + x2 = 14.66666667 Factor a perfect square on the left side: (x + 1)(x + 1) = 14.66666667 Calculate the square root of the right side: 3.829708431 Break this problem into two subproblems by setting (x + 1) equal to 3.829708431 and -3.829708431.Subproblem 1
x + 1 = 3.829708431 Simplifying x + 1 = 3.829708431 Reorder the terms: 1 + x = 3.829708431 Solving 1 + x = 3.829708431 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + x = 3.829708431 + -1 Combine like terms: 1 + -1 = 0 0 + x = 3.829708431 + -1 x = 3.829708431 + -1 Combine like terms: 3.829708431 + -1 = 2.829708431 x = 2.829708431 Simplifying x = 2.829708431Subproblem 2
x + 1 = -3.829708431 Simplifying x + 1 = -3.829708431 Reorder the terms: 1 + x = -3.829708431 Solving 1 + x = -3.829708431 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + x = -3.829708431 + -1 Combine like terms: 1 + -1 = 0 0 + x = -3.829708431 + -1 x = -3.829708431 + -1 Combine like terms: -3.829708431 + -1 = -4.829708431 x = -4.829708431 Simplifying x = -4.829708431Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.829708431, -4.829708431}Solution
x = {2.829708431, -4.829708431}
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