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Simplifying 9x2 + 27x + 1 = 0 Reorder the terms: 1 + 27x + 9x2 = 0 Solving 1 + 27x + 9x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. 0.1111111111 + 3x + x2 = 0 Move the constant term to the right: Add '-0.1111111111' to each side of the equation. 0.1111111111 + 3x + -0.1111111111 + x2 = 0 + -0.1111111111 Reorder the terms: 0.1111111111 + -0.1111111111 + 3x + x2 = 0 + -0.1111111111 Combine like terms: 0.1111111111 + -0.1111111111 = 0.0000000000 0.0000000000 + 3x + x2 = 0 + -0.1111111111 3x + x2 = 0 + -0.1111111111 Combine like terms: 0 + -0.1111111111 = -0.1111111111 3x + x2 = -0.1111111111 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 = -0.1111111111 + 2.25 Reorder the terms: 2.25 + 3x + x2 = -0.1111111111 + 2.25 Combine like terms: -0.1111111111 + 2.25 = 2.1388888889 2.25 + 3x + x2 = 2.1388888889 Factor a perfect square on the left side: (x + 1.5)(x + 1.5) = 2.1388888889 Calculate the square root of the right side: 1.462494065 Break this problem into two subproblems by setting (x + 1.5) equal to 1.462494065 and -1.462494065.Subproblem 1
x + 1.5 = 1.462494065 Simplifying x + 1.5 = 1.462494065 Reorder the terms: 1.5 + x = 1.462494065 Solving 1.5 + x = 1.462494065 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 = 1.462494065 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = 1.462494065 + -1.5 x = 1.462494065 + -1.5 Combine like terms: 1.462494065 + -1.5 = -0.037505935 x = -0.037505935 Simplifying x = -0.037505935Subproblem 2
x + 1.5 = -1.462494065 Simplifying x + 1.5 = -1.462494065 Reorder the terms: 1.5 + x = -1.462494065 Solving 1.5 + x = -1.462494065 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 = -1.462494065 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = -1.462494065 + -1.5 x = -1.462494065 + -1.5 Combine like terms: -1.462494065 + -1.5 = -2.962494065 x = -2.962494065 Simplifying x = -2.962494065Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.037505935, -2.962494065}
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