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Simplifying 12x2 + 3x = 27 Reorder the terms: 3x + 12x2 = 27 Solving 3x + 12x2 = 27 Solving for variable 'x'. Reorder the terms: -27 + 3x + 12x2 = 27 + -27 Combine like terms: 27 + -27 = 0 -27 + 3x + 12x2 = 0 Factor out the Greatest Common Factor (GCF), '3'. 3(-9 + x + 4x2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-9 + x + 4x2)' equal to zero and attempt to solve: Simplifying -9 + x + 4x2 = 0 Solving -9 + x + 4x2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -2.25 + 0.25x + x2 = 0 Move the constant term to the right: Add '2.25' to each side of the equation. -2.25 + 0.25x + 2.25 + x2 = 0 + 2.25 Reorder the terms: -2.25 + 2.25 + 0.25x + x2 = 0 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + 0.25x + x2 = 0 + 2.25 0.25x + x2 = 0 + 2.25 Combine like terms: 0 + 2.25 = 2.25 0.25x + x2 = 2.25 The x term is x. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. 0.25x + 0.25 + x2 = 2.25 + 0.25 Reorder the terms: 0.25 + 0.25x + x2 = 2.25 + 0.25 Combine like terms: 2.25 + 0.25 = 2.5 0.25 + 0.25x + x2 = 2.5 Factor a perfect square on the left side: (x + 0.5)(x + 0.5) = 2.5 Calculate the square root of the right side: 1.58113883 Break this problem into two subproblems by setting (x + 0.5) equal to 1.58113883 and -1.58113883.Subproblem 1
x + 0.5 = 1.58113883 Simplifying x + 0.5 = 1.58113883 Reorder the terms: 0.5 + x = 1.58113883 Solving 0.5 + x = 1.58113883 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = 1.58113883 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 1.58113883 + -0.5 x = 1.58113883 + -0.5 Combine like terms: 1.58113883 + -0.5 = 1.08113883 x = 1.08113883 Simplifying x = 1.08113883Subproblem 2
x + 0.5 = -1.58113883 Simplifying x + 0.5 = -1.58113883 Reorder the terms: 0.5 + x = -1.58113883 Solving 0.5 + x = -1.58113883 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = -1.58113883 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -1.58113883 + -0.5 x = -1.58113883 + -0.5 Combine like terms: -1.58113883 + -0.5 = -2.08113883 x = -2.08113883 Simplifying x = -2.08113883Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.08113883, -2.08113883}Solution
x = {1.08113883, -2.08113883}
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