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Simplifying 6x2 + 27x + -234 = 0 Reorder the terms: -234 + 27x + 6x2 = 0 Solving -234 + 27x + 6x2 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '3'. 3(-78 + 9x + 2x2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-78 + 9x + 2x2)' equal to zero and attempt to solve: Simplifying -78 + 9x + 2x2 = 0 Solving -78 + 9x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -39 + 4.5x + x2 = 0 Move the constant term to the right: Add '39' to each side of the equation. -39 + 4.5x + 39 + x2 = 0 + 39 Reorder the terms: -39 + 39 + 4.5x + x2 = 0 + 39 Combine like terms: -39 + 39 = 0 0 + 4.5x + x2 = 0 + 39 4.5x + x2 = 0 + 39 Combine like terms: 0 + 39 = 39 4.5x + x2 = 39 The x term is 4.5x. Take half its coefficient (2.25). Square it (5.0625) and add it to both sides. Add '5.0625' to each side of the equation. 4.5x + 5.0625 + x2 = 39 + 5.0625 Reorder the terms: 5.0625 + 4.5x + x2 = 39 + 5.0625 Combine like terms: 39 + 5.0625 = 44.0625 5.0625 + 4.5x + x2 = 44.0625 Factor a perfect square on the left side: (x + 2.25)(x + 2.25) = 44.0625 Calculate the square root of the right side: 6.637959024 Break this problem into two subproblems by setting (x + 2.25) equal to 6.637959024 and -6.637959024.Subproblem 1
x + 2.25 = 6.637959024 Simplifying x + 2.25 = 6.637959024 Reorder the terms: 2.25 + x = 6.637959024 Solving 2.25 + x = 6.637959024 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.25' to each side of the equation. 2.25 + -2.25 + x = 6.637959024 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + x = 6.637959024 + -2.25 x = 6.637959024 + -2.25 Combine like terms: 6.637959024 + -2.25 = 4.387959024 x = 4.387959024 Simplifying x = 4.387959024Subproblem 2
x + 2.25 = -6.637959024 Simplifying x + 2.25 = -6.637959024 Reorder the terms: 2.25 + x = -6.637959024 Solving 2.25 + x = -6.637959024 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.25' to each side of the equation. 2.25 + -2.25 + x = -6.637959024 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + x = -6.637959024 + -2.25 x = -6.637959024 + -2.25 Combine like terms: -6.637959024 + -2.25 = -8.887959024 x = -8.887959024 Simplifying x = -8.887959024Solution
The solution to the problem is based on the solutions from the subproblems. x = {4.387959024, -8.887959024}Solution
x = {4.387959024, -8.887959024}
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