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Simplifying 12x2 + -39x + 24 = 0 Reorder the terms: 24 + -39x + 12x2 = 0 Solving 24 + -39x + 12x2 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '3'. 3(8 + -13x + 4x2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(8 + -13x + 4x2)' equal to zero and attempt to solve: Simplifying 8 + -13x + 4x2 = 0 Solving 8 + -13x + 4x2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 2 + -3.25x + x2 = 0 Move the constant term to the right: Add '-2' to each side of the equation. 2 + -3.25x + -2 + x2 = 0 + -2 Reorder the terms: 2 + -2 + -3.25x + x2 = 0 + -2 Combine like terms: 2 + -2 = 0 0 + -3.25x + x2 = 0 + -2 -3.25x + x2 = 0 + -2 Combine like terms: 0 + -2 = -2 -3.25x + x2 = -2 The x term is -3.25x. Take half its coefficient (-1.625). Square it (2.640625) and add it to both sides. Add '2.640625' to each side of the equation. -3.25x + 2.640625 + x2 = -2 + 2.640625 Reorder the terms: 2.640625 + -3.25x + x2 = -2 + 2.640625 Combine like terms: -2 + 2.640625 = 0.640625 2.640625 + -3.25x + x2 = 0.640625 Factor a perfect square on the left side: (x + -1.625)(x + -1.625) = 0.640625 Calculate the square root of the right side: 0.80039053 Break this problem into two subproblems by setting (x + -1.625) equal to 0.80039053 and -0.80039053.Subproblem 1
x + -1.625 = 0.80039053 Simplifying x + -1.625 = 0.80039053 Reorder the terms: -1.625 + x = 0.80039053 Solving -1.625 + x = 0.80039053 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.625' to each side of the equation. -1.625 + 1.625 + x = 0.80039053 + 1.625 Combine like terms: -1.625 + 1.625 = 0.000 0.000 + x = 0.80039053 + 1.625 x = 0.80039053 + 1.625 Combine like terms: 0.80039053 + 1.625 = 2.42539053 x = 2.42539053 Simplifying x = 2.42539053Subproblem 2
x + -1.625 = -0.80039053 Simplifying x + -1.625 = -0.80039053 Reorder the terms: -1.625 + x = -0.80039053 Solving -1.625 + x = -0.80039053 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.625' to each side of the equation. -1.625 + 1.625 + x = -0.80039053 + 1.625 Combine like terms: -1.625 + 1.625 = 0.000 0.000 + x = -0.80039053 + 1.625 x = -0.80039053 + 1.625 Combine like terms: -0.80039053 + 1.625 = 0.82460947 x = 0.82460947 Simplifying x = 0.82460947Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.42539053, 0.82460947}Solution
x = {2.42539053, 0.82460947}
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