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Simplifying -27x2 + 54x + 225 = 0 Reorder the terms: 225 + 54x + -27x2 = 0 Solving 225 + 54x + -27x2 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '9'. 9(25 + 6x + -3x2) = 0 Ignore the factor 9.Subproblem 1
Set the factor '(25 + 6x + -3x2)' equal to zero and attempt to solve: Simplifying 25 + 6x + -3x2 = 0 Solving 25 + 6x + -3x2 = 0 Begin completing the square. Divide all terms by -3 the coefficient of the squared term: Divide each side by '-3'. -8.333333333 + -2x + x2 = 0 Move the constant term to the right: Add '8.333333333' to each side of the equation. -8.333333333 + -2x + 8.333333333 + x2 = 0 + 8.333333333 Reorder the terms: -8.333333333 + 8.333333333 + -2x + x2 = 0 + 8.333333333 Combine like terms: -8.333333333 + 8.333333333 = 0.000000000 0.000000000 + -2x + x2 = 0 + 8.333333333 -2x + x2 = 0 + 8.333333333 Combine like terms: 0 + 8.333333333 = 8.333333333 -2x + x2 = 8.333333333 The x term is -2x. Take half its coefficient (-1). Square it (1) and add it to both sides. Add '1' to each side of the equation. -2x + 1 + x2 = 8.333333333 + 1 Reorder the terms: 1 + -2x + x2 = 8.333333333 + 1 Combine like terms: 8.333333333 + 1 = 9.333333333 1 + -2x + x2 = 9.333333333 Factor a perfect square on the left side: (x + -1)(x + -1) = 9.333333333 Calculate the square root of the right side: 3.055050463 Break this problem into two subproblems by setting (x + -1) equal to 3.055050463 and -3.055050463.Subproblem 1
x + -1 = 3.055050463 Simplifying x + -1 = 3.055050463 Reorder the terms: -1 + x = 3.055050463 Solving -1 + x = 3.055050463 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + x = 3.055050463 + 1 Combine like terms: -1 + 1 = 0 0 + x = 3.055050463 + 1 x = 3.055050463 + 1 Combine like terms: 3.055050463 + 1 = 4.055050463 x = 4.055050463 Simplifying x = 4.055050463Subproblem 2
x + -1 = -3.055050463 Simplifying x + -1 = -3.055050463 Reorder the terms: -1 + x = -3.055050463 Solving -1 + x = -3.055050463 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + x = -3.055050463 + 1 Combine like terms: -1 + 1 = 0 0 + x = -3.055050463 + 1 x = -3.055050463 + 1 Combine like terms: -3.055050463 + 1 = -2.055050463 x = -2.055050463 Simplifying x = -2.055050463Solution
The solution to the problem is based on the solutions from the subproblems. x = {4.055050463, -2.055050463}Solution
x = {4.055050463, -2.055050463}
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