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Simplifying 10 = -6x2 + 24x + 20 Reorder the terms: 10 = 20 + 24x + -6x2 Solving 10 = 20 + 24x + -6x2 Solving for variable 'x'. Combine like terms: 10 + -20 = -10 -10 + -24x + 6x2 = 20 + 24x + -6x2 + -20 + -24x + 6x2 Reorder the terms: -10 + -24x + 6x2 = 20 + -20 + 24x + -24x + -6x2 + 6x2 Combine like terms: 20 + -20 = 0 -10 + -24x + 6x2 = 0 + 24x + -24x + -6x2 + 6x2 -10 + -24x + 6x2 = 24x + -24x + -6x2 + 6x2 Combine like terms: 24x + -24x = 0 -10 + -24x + 6x2 = 0 + -6x2 + 6x2 -10 + -24x + 6x2 = -6x2 + 6x2 Combine like terms: -6x2 + 6x2 = 0 -10 + -24x + 6x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-5 + -12x + 3x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-5 + -12x + 3x2)' equal to zero and attempt to solve: Simplifying -5 + -12x + 3x2 = 0 Solving -5 + -12x + 3x2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -1.666666667 + -4x + x2 = 0 Move the constant term to the right: Add '1.666666667' to each side of the equation. -1.666666667 + -4x + 1.666666667 + x2 = 0 + 1.666666667 Reorder the terms: -1.666666667 + 1.666666667 + -4x + x2 = 0 + 1.666666667 Combine like terms: -1.666666667 + 1.666666667 = 0.000000000 0.000000000 + -4x + x2 = 0 + 1.666666667 -4x + x2 = 0 + 1.666666667 Combine like terms: 0 + 1.666666667 = 1.666666667 -4x + x2 = 1.666666667 The x term is -4x. Take half its coefficient (-2). Square it (4) and add it to both sides. Add '4' to each side of the equation. -4x + 4 + x2 = 1.666666667 + 4 Reorder the terms: 4 + -4x + x2 = 1.666666667 + 4 Combine like terms: 1.666666667 + 4 = 5.666666667 4 + -4x + x2 = 5.666666667 Factor a perfect square on the left side: (x + -2)(x + -2) = 5.666666667 Calculate the square root of the right side: 2.380476143 Break this problem into two subproblems by setting (x + -2) equal to 2.380476143 and -2.380476143.Subproblem 1
x + -2 = 2.380476143 Simplifying x + -2 = 2.380476143 Reorder the terms: -2 + x = 2.380476143 Solving -2 + x = 2.380476143 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + x = 2.380476143 + 2 Combine like terms: -2 + 2 = 0 0 + x = 2.380476143 + 2 x = 2.380476143 + 2 Combine like terms: 2.380476143 + 2 = 4.380476143 x = 4.380476143 Simplifying x = 4.380476143Subproblem 2
x + -2 = -2.380476143 Simplifying x + -2 = -2.380476143 Reorder the terms: -2 + x = -2.380476143 Solving -2 + x = -2.380476143 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + x = -2.380476143 + 2 Combine like terms: -2 + 2 = 0 0 + x = -2.380476143 + 2 x = -2.380476143 + 2 Combine like terms: -2.380476143 + 2 = -0.380476143 x = -0.380476143 Simplifying x = -0.380476143Solution
The solution to the problem is based on the solutions from the subproblems. x = {4.380476143, -0.380476143}Solution
x = {4.380476143, -0.380476143}
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