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Simplifying 0 = 6x2 + -14x + -24 Reorder the terms: 0 = -24 + -14x + 6x2 Solving 0 = -24 + -14x + 6x2 Solving for variable 'x'. Combine like terms: 0 + 24 = 24 24 + 14x + -6x2 = -24 + -14x + 6x2 + 24 + 14x + -6x2 Reorder the terms: 24 + 14x + -6x2 = -24 + 24 + -14x + 14x + 6x2 + -6x2 Combine like terms: -24 + 24 = 0 24 + 14x + -6x2 = 0 + -14x + 14x + 6x2 + -6x2 24 + 14x + -6x2 = -14x + 14x + 6x2 + -6x2 Combine like terms: -14x + 14x = 0 24 + 14x + -6x2 = 0 + 6x2 + -6x2 24 + 14x + -6x2 = 6x2 + -6x2 Combine like terms: 6x2 + -6x2 = 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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