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Simplifying 12x2 + 36x + 8 = 0 Reorder the terms: 8 + 36x + 12x2 = 0 Solving 8 + 36x + 12x2 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '4'. 4(2 + 9x + 3x2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(2 + 9x + 3x2)' equal to zero and attempt to solve: Simplifying 2 + 9x + 3x2 = 0 Solving 2 + 9x + 3x2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 0.6666666667 + 3x + x2 = 0 Move the constant term to the right: Add '-0.6666666667' to each side of the equation. 0.6666666667 + 3x + -0.6666666667 + x2 = 0 + -0.6666666667 Reorder the terms: 0.6666666667 + -0.6666666667 + 3x + x2 = 0 + -0.6666666667 Combine like terms: 0.6666666667 + -0.6666666667 = 0.0000000000 0.0000000000 + 3x + x2 = 0 + -0.6666666667 3x + x2 = 0 + -0.6666666667 Combine like terms: 0 + -0.6666666667 = -0.6666666667 3x + x2 = -0.6666666667 The x term is 3x. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3x + 2.25 + x2 = -0.6666666667 + 2.25 Reorder the terms: 2.25 + 3x + x2 = -0.6666666667 + 2.25 Combine like terms: -0.6666666667 + 2.25 = 1.5833333333 2.25 + 3x + x2 = 1.5833333333 Factor a perfect square on the left side: (x + 1.5)(x + 1.5) = 1.5833333333 Calculate the square root of the right side: 1.258305739 Break this problem into two subproblems by setting (x + 1.5) equal to 1.258305739 and -1.258305739.Subproblem 1
x + 1.5 = 1.258305739 Simplifying x + 1.5 = 1.258305739 Reorder the terms: 1.5 + x = 1.258305739 Solving 1.5 + x = 1.258305739 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + x = 1.258305739 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = 1.258305739 + -1.5 x = 1.258305739 + -1.5 Combine like terms: 1.258305739 + -1.5 = -0.241694261 x = -0.241694261 Simplifying x = -0.241694261Subproblem 2
x + 1.5 = -1.258305739 Simplifying x + 1.5 = -1.258305739 Reorder the terms: 1.5 + x = -1.258305739 Solving 1.5 + x = -1.258305739 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + x = -1.258305739 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = -1.258305739 + -1.5 x = -1.258305739 + -1.5 Combine like terms: -1.258305739 + -1.5 = -2.758305739 x = -2.758305739 Simplifying x = -2.758305739Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.241694261, -2.758305739}Solution
x = {-0.241694261, -2.758305739}
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