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Simplifying 23 = 2x2 + -12x + 25 Reorder the terms: 23 = 25 + -12x + 2x2 Solving 23 = 25 + -12x + 2x2 Solving for variable 'x'. Combine like terms: 23 + -25 = -2 -2 + 12x + -2x2 = 25 + -12x + 2x2 + -25 + 12x + -2x2 Reorder the terms: -2 + 12x + -2x2 = 25 + -25 + -12x + 12x + 2x2 + -2x2 Combine like terms: 25 + -25 = 0 -2 + 12x + -2x2 = 0 + -12x + 12x + 2x2 + -2x2 -2 + 12x + -2x2 = -12x + 12x + 2x2 + -2x2 Combine like terms: -12x + 12x = 0 -2 + 12x + -2x2 = 0 + 2x2 + -2x2 -2 + 12x + -2x2 = 2x2 + -2x2 Combine like terms: 2x2 + -2x2 = 0 -2 + 12x + -2x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-1 + 6x + -1x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-1 + 6x + -1x2)' equal to zero and attempt to solve: Simplifying -1 + 6x + -1x2 = 0 Solving -1 + 6x + -1x2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. 1 + -6x + x2 = 0 Move the constant term to the right: Add '-1' to each side of the equation. 1 + -6x + -1 + x2 = 0 + -1 Reorder the terms: 1 + -1 + -6x + x2 = 0 + -1 Combine like terms: 1 + -1 = 0 0 + -6x + x2 = 0 + -1 -6x + x2 = 0 + -1 Combine like terms: 0 + -1 = -1 -6x + x2 = -1 The x term is -6x. Take half its coefficient (-3). Square it (9) and add it to both sides. Add '9' to each side of the equation. -6x + 9 + x2 = -1 + 9 Reorder the terms: 9 + -6x + x2 = -1 + 9 Combine like terms: -1 + 9 = 8 9 + -6x + x2 = 8 Factor a perfect square on the left side: (x + -3)(x + -3) = 8 Calculate the square root of the right side: 2.828427125 Break this problem into two subproblems by setting (x + -3) equal to 2.828427125 and -2.828427125.Subproblem 1
x + -3 = 2.828427125 Simplifying x + -3 = 2.828427125 Reorder the terms: -3 + x = 2.828427125 Solving -3 + x = 2.828427125 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + x = 2.828427125 + 3 Combine like terms: -3 + 3 = 0 0 + x = 2.828427125 + 3 x = 2.828427125 + 3 Combine like terms: 2.828427125 + 3 = 5.828427125 x = 5.828427125 Simplifying x = 5.828427125Subproblem 2
x + -3 = -2.828427125 Simplifying x + -3 = -2.828427125 Reorder the terms: -3 + x = -2.828427125 Solving -3 + x = -2.828427125 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + x = -2.828427125 + 3 Combine like terms: -3 + 3 = 0 0 + x = -2.828427125 + 3 x = -2.828427125 + 3 Combine like terms: -2.828427125 + 3 = 0.171572875 x = 0.171572875 Simplifying x = 0.171572875Solution
The solution to the problem is based on the solutions from the subproblems. x = {5.828427125, 0.171572875}Solution
x = {5.828427125, 0.171572875}
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