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Simplifying 9x2 + 27x = 7 Reorder the terms: 27x + 9x2 = 7 Solving 27x + 9x2 = 7 Solving for variable 'x'. Reorder the terms: -7 + 27x + 9x2 = 7 + -7 Combine like terms: 7 + -7 = 0 -7 + 27x + 9x2 = 0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -0.7777777778 + 3x + x2 = 0 Move the constant term to the right: Add '0.7777777778' to each side of the equation. -0.7777777778 + 3x + 0.7777777778 + x2 = 0 + 0.7777777778 Reorder the terms: -0.7777777778 + 0.7777777778 + 3x + x2 = 0 + 0.7777777778 Combine like terms: -0.7777777778 + 0.7777777778 = 0.0000000000 0.0000000000 + 3x + x2 = 0 + 0.7777777778 3x + x2 = 0 + 0.7777777778 Combine like terms: 0 + 0.7777777778 = 0.7777777778 3x + x2 = 0.7777777778 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 = 0.7777777778 + 2.25 Reorder the terms: 2.25 + 3x + x2 = 0.7777777778 + 2.25 Combine like terms: 0.7777777778 + 2.25 = 3.0277777778 2.25 + 3x + x2 = 3.0277777778 Factor a perfect square on the left side: (x + 1.5)(x + 1.5) = 3.0277777778 Calculate the square root of the right side: 1.740051085 Break this problem into two subproblems by setting (x + 1.5) equal to 1.740051085 and -1.740051085.Subproblem 1
x + 1.5 = 1.740051085 Simplifying x + 1.5 = 1.740051085 Reorder the terms: 1.5 + x = 1.740051085 Solving 1.5 + x = 1.740051085 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 = 1.740051085 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = 1.740051085 + -1.5 x = 1.740051085 + -1.5 Combine like terms: 1.740051085 + -1.5 = 0.240051085 x = 0.240051085 Simplifying x = 0.240051085Subproblem 2
x + 1.5 = -1.740051085 Simplifying x + 1.5 = -1.740051085 Reorder the terms: 1.5 + x = -1.740051085 Solving 1.5 + x = -1.740051085 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 = -1.740051085 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = -1.740051085 + -1.5 x = -1.740051085 + -1.5 Combine like terms: -1.740051085 + -1.5 = -3.240051085 x = -3.240051085 Simplifying x = -3.240051085Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.240051085, -3.240051085}
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