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Simplifying 9x2 + 27x = 306 Reorder the terms: 27x + 9x2 = 306 Solving 27x + 9x2 = 306 Solving for variable 'x'. Reorder the terms: -306 + 27x + 9x2 = 306 + -306 Combine like terms: 306 + -306 = 0 -306 + 27x + 9x2 = 0 Factor out the Greatest Common Factor (GCF), '9'. 9(-34 + 3x + x2) = 0 Ignore the factor 9.Subproblem 1
Set the factor '(-34 + 3x + x2)' equal to zero and attempt to solve: Simplifying -34 + 3x + x2 = 0 Solving -34 + 3x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '34' to each side of the equation. -34 + 3x + 34 + x2 = 0 + 34 Reorder the terms: -34 + 34 + 3x + x2 = 0 + 34 Combine like terms: -34 + 34 = 0 0 + 3x + x2 = 0 + 34 3x + x2 = 0 + 34 Combine like terms: 0 + 34 = 34 3x + x2 = 34 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 = 34 + 2.25 Reorder the terms: 2.25 + 3x + x2 = 34 + 2.25 Combine like terms: 34 + 2.25 = 36.25 2.25 + 3x + x2 = 36.25 Factor a perfect square on the left side: (x + 1.5)(x + 1.5) = 36.25 Calculate the square root of the right side: 6.020797289 Break this problem into two subproblems by setting (x + 1.5) equal to 6.020797289 and -6.020797289.Subproblem 1
x + 1.5 = 6.020797289 Simplifying x + 1.5 = 6.020797289 Reorder the terms: 1.5 + x = 6.020797289 Solving 1.5 + x = 6.020797289 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 = 6.020797289 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = 6.020797289 + -1.5 x = 6.020797289 + -1.5 Combine like terms: 6.020797289 + -1.5 = 4.520797289 x = 4.520797289 Simplifying x = 4.520797289Subproblem 2
x + 1.5 = -6.020797289 Simplifying x + 1.5 = -6.020797289 Reorder the terms: 1.5 + x = -6.020797289 Solving 1.5 + x = -6.020797289 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 = -6.020797289 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = -6.020797289 + -1.5 x = -6.020797289 + -1.5 Combine like terms: -6.020797289 + -1.5 = -7.520797289 x = -7.520797289 Simplifying x = -7.520797289Solution
The solution to the problem is based on the solutions from the subproblems. x = {4.520797289, -7.520797289}Solution
x = {4.520797289, -7.520797289}
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