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Simplifying 6x2 = 14x + 24 Reorder the terms: 6x2 = 24 + 14x Solving 6x2 = 24 + 14x Solving for variable 'x'. Reorder the terms: -24 + -14x + 6x2 = 24 + 14x + -24 + -14x Reorder the terms: -24 + -14x + 6x2 = 24 + -24 + 14x + -14x Combine like terms: 24 + -24 = 0 -24 + -14x + 6x2 = 0 + 14x + -14x -24 + -14x + 6x2 = 14x + -14x Combine like terms: 14x + -14x = 0 -24 + -14x + 6x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-12 + -7x + 3x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-12 + -7x + 3x2)' equal to zero and attempt to solve: Simplifying -12 + -7x + 3x2 = 0 Solving -12 + -7x + 3x2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -4 + -2.333333333x + x2 = 0 Move the constant term to the right: Add '4' to each side of the equation. -4 + -2.333333333x + 4 + x2 = 0 + 4 Reorder the terms: -4 + 4 + -2.333333333x + x2 = 0 + 4 Combine like terms: -4 + 4 = 0 0 + -2.333333333x + x2 = 0 + 4 -2.333333333x + x2 = 0 + 4 Combine like terms: 0 + 4 = 4 -2.333333333x + x2 = 4 The x term is -2.333333333x. Take half its coefficient (-1.166666667). Square it (1.361111112) and add it to both sides. Add '1.361111112' to each side of the equation. -2.333333333x + 1.361111112 + x2 = 4 + 1.361111112 Reorder the terms: 1.361111112 + -2.333333333x + x2 = 4 + 1.361111112 Combine like terms: 4 + 1.361111112 = 5.361111112 1.361111112 + -2.333333333x + x2 = 5.361111112 Factor a perfect square on the left side: (x + -1.166666667)(x + -1.166666667) = 5.361111112 Calculate the square root of the right side: 2.315407332 Break this problem into two subproblems by setting (x + -1.166666667) equal to 2.315407332 and -2.315407332.Subproblem 1
x + -1.166666667 = 2.315407332 Simplifying x + -1.166666667 = 2.315407332 Reorder the terms: -1.166666667 + x = 2.315407332 Solving -1.166666667 + x = 2.315407332 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.166666667' to each side of the equation. -1.166666667 + 1.166666667 + x = 2.315407332 + 1.166666667 Combine like terms: -1.166666667 + 1.166666667 = 0.000000000 0.000000000 + x = 2.315407332 + 1.166666667 x = 2.315407332 + 1.166666667 Combine like terms: 2.315407332 + 1.166666667 = 3.482073999 x = 3.482073999 Simplifying x = 3.482073999Subproblem 2
x + -1.166666667 = -2.315407332 Simplifying x + -1.166666667 = -2.315407332 Reorder the terms: -1.166666667 + x = -2.315407332 Solving -1.166666667 + x = -2.315407332 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.166666667' to each side of the equation. -1.166666667 + 1.166666667 + x = -2.315407332 + 1.166666667 Combine like terms: -1.166666667 + 1.166666667 = 0.000000000 0.000000000 + x = -2.315407332 + 1.166666667 x = -2.315407332 + 1.166666667 Combine like terms: -2.315407332 + 1.166666667 = -1.148740665 x = -1.148740665 Simplifying x = -1.148740665Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.482073999, -1.148740665}Solution
x = {3.482073999, -1.148740665}
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