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Simplifying 0 = 12x2 + -24x + 8 Reorder the terms: 0 = 8 + -24x + 12x2 Solving 0 = 8 + -24x + 12x2 Solving for variable 'x'. Combine like terms: 0 + -8 = -8 -8 + 24x + -12x2 = 8 + -24x + 12x2 + -8 + 24x + -12x2 Reorder the terms: -8 + 24x + -12x2 = 8 + -8 + -24x + 24x + 12x2 + -12x2 Combine like terms: 8 + -8 = 0 -8 + 24x + -12x2 = 0 + -24x + 24x + 12x2 + -12x2 -8 + 24x + -12x2 = -24x + 24x + 12x2 + -12x2 Combine like terms: -24x + 24x = 0 -8 + 24x + -12x2 = 0 + 12x2 + -12x2 -8 + 24x + -12x2 = 12x2 + -12x2 Combine like terms: 12x2 + -12x2 = 0 -8 + 24x + -12x2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(-2 + 6x + -3x2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-2 + 6x + -3x2)' equal to zero and attempt to solve: Simplifying -2 + 6x + -3x2 = 0 Solving -2 + 6x + -3x2 = 0 Begin completing the square. Divide all terms by -3 the coefficient of the squared term: Divide each side by '-3'. 0.6666666667 + -2x + x2 = 0 Move the constant term to the right: Add '-0.6666666667' to each side of the equation. 0.6666666667 + -2x + -0.6666666667 + x2 = 0 + -0.6666666667 Reorder the terms: 0.6666666667 + -0.6666666667 + -2x + x2 = 0 + -0.6666666667 Combine like terms: 0.6666666667 + -0.6666666667 = 0.0000000000 0.0000000000 + -2x + x2 = 0 + -0.6666666667 -2x + x2 = 0 + -0.6666666667 Combine like terms: 0 + -0.6666666667 = -0.6666666667 -2x + x2 = -0.6666666667 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 = -0.6666666667 + 1 Reorder the terms: 1 + -2x + x2 = -0.6666666667 + 1 Combine like terms: -0.6666666667 + 1 = 0.3333333333 1 + -2x + x2 = 0.3333333333 Factor a perfect square on the left side: (x + -1)(x + -1) = 0.3333333333 Calculate the square root of the right side: 0.577350269 Break this problem into two subproblems by setting (x + -1) equal to 0.577350269 and -0.577350269.Subproblem 1
x + -1 = 0.577350269 Simplifying x + -1 = 0.577350269 Reorder the terms: -1 + x = 0.577350269 Solving -1 + x = 0.577350269 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 = 0.577350269 + 1 Combine like terms: -1 + 1 = 0 0 + x = 0.577350269 + 1 x = 0.577350269 + 1 Combine like terms: 0.577350269 + 1 = 1.577350269 x = 1.577350269 Simplifying x = 1.577350269Subproblem 2
x + -1 = -0.577350269 Simplifying x + -1 = -0.577350269 Reorder the terms: -1 + x = -0.577350269 Solving -1 + x = -0.577350269 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 = -0.577350269 + 1 Combine like terms: -1 + 1 = 0 0 + x = -0.577350269 + 1 x = -0.577350269 + 1 Combine like terms: -0.577350269 + 1 = 0.422649731 x = 0.422649731 Simplifying x = 0.422649731Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.577350269, 0.422649731}Solution
x = {1.577350269, 0.422649731}
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