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Simplifying 0 = 12x2 + 24x + -14 Reorder the terms: 0 = -14 + 24x + 12x2 Solving 0 = -14 + 24x + 12x2 Solving for variable 'x'. Combine like terms: 0 + 14 = 14 14 + -24x + -12x2 = -14 + 24x + 12x2 + 14 + -24x + -12x2 Reorder the terms: 14 + -24x + -12x2 = -14 + 14 + 24x + -24x + 12x2 + -12x2 Combine like terms: -14 + 14 = 0 14 + -24x + -12x2 = 0 + 24x + -24x + 12x2 + -12x2 14 + -24x + -12x2 = 24x + -24x + 12x2 + -12x2 Combine like terms: 24x + -24x = 0 14 + -24x + -12x2 = 0 + 12x2 + -12x2 14 + -24x + -12x2 = 12x2 + -12x2 Combine like terms: 12x2 + -12x2 = 0 14 + -24x + -12x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(7 + -12x + -6x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(7 + -12x + -6x2)' equal to zero and attempt to solve: Simplifying 7 + -12x + -6x2 = 0 Solving 7 + -12x + -6x2 = 0 Begin completing the square. Divide all terms by -6 the coefficient of the squared term: Divide each side by '-6'. -1.166666667 + 2x + x2 = 0 Move the constant term to the right: Add '1.166666667' to each side of the equation. -1.166666667 + 2x + 1.166666667 + x2 = 0 + 1.166666667 Reorder the terms: -1.166666667 + 1.166666667 + 2x + x2 = 0 + 1.166666667 Combine like terms: -1.166666667 + 1.166666667 = 0.000000000 0.000000000 + 2x + x2 = 0 + 1.166666667 2x + x2 = 0 + 1.166666667 Combine like terms: 0 + 1.166666667 = 1.166666667 2x + x2 = 1.166666667 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 = 1.166666667 + 1 Reorder the terms: 1 + 2x + x2 = 1.166666667 + 1 Combine like terms: 1.166666667 + 1 = 2.166666667 1 + 2x + x2 = 2.166666667 Factor a perfect square on the left side: (x + 1)(x + 1) = 2.166666667 Calculate the square root of the right side: 1.471960145 Break this problem into two subproblems by setting (x + 1) equal to 1.471960145 and -1.471960145.Subproblem 1
x + 1 = 1.471960145 Simplifying x + 1 = 1.471960145 Reorder the terms: 1 + x = 1.471960145 Solving 1 + x = 1.471960145 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 = 1.471960145 + -1 Combine like terms: 1 + -1 = 0 0 + x = 1.471960145 + -1 x = 1.471960145 + -1 Combine like terms: 1.471960145 + -1 = 0.471960145 x = 0.471960145 Simplifying x = 0.471960145Subproblem 2
x + 1 = -1.471960145 Simplifying x + 1 = -1.471960145 Reorder the terms: 1 + x = -1.471960145 Solving 1 + x = -1.471960145 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 = -1.471960145 + -1 Combine like terms: 1 + -1 = 0 0 + x = -1.471960145 + -1 x = -1.471960145 + -1 Combine like terms: -1.471960145 + -1 = -2.471960145 x = -2.471960145 Simplifying x = -2.471960145Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.471960145, -2.471960145}Solution
x = {0.471960145, -2.471960145}
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