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Simplifying 18x2 + -40x + -27 = 0 Reorder the terms: -27 + -40x + 18x2 = 0 Solving -27 + -40x + 18x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 18 the coefficient of the squared term: Divide each side by '18'. -1.5 + -2.222222222x + x2 = 0 Move the constant term to the right: Add '1.5' to each side of the equation. -1.5 + -2.222222222x + 1.5 + x2 = 0 + 1.5 Reorder the terms: -1.5 + 1.5 + -2.222222222x + x2 = 0 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + -2.222222222x + x2 = 0 + 1.5 -2.222222222x + x2 = 0 + 1.5 Combine like terms: 0 + 1.5 = 1.5 -2.222222222x + x2 = 1.5 The x term is -2.222222222x. Take half its coefficient (-1.111111111). Square it (1.234567901) and add it to both sides. Add '1.234567901' to each side of the equation. -2.222222222x + 1.234567901 + x2 = 1.5 + 1.234567901 Reorder the terms: 1.234567901 + -2.222222222x + x2 = 1.5 + 1.234567901 Combine like terms: 1.5 + 1.234567901 = 2.734567901 1.234567901 + -2.222222222x + x2 = 2.734567901 Factor a perfect square on the left side: (x + -1.111111111)(x + -1.111111111) = 2.734567901 Calculate the square root of the right side: 1.653652896 Break this problem into two subproblems by setting (x + -1.111111111) equal to 1.653652896 and -1.653652896.Subproblem 1
x + -1.111111111 = 1.653652896 Simplifying x + -1.111111111 = 1.653652896 Reorder the terms: -1.111111111 + x = 1.653652896 Solving -1.111111111 + x = 1.653652896 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.111111111' to each side of the equation. -1.111111111 + 1.111111111 + x = 1.653652896 + 1.111111111 Combine like terms: -1.111111111 + 1.111111111 = 0.000000000 0.000000000 + x = 1.653652896 + 1.111111111 x = 1.653652896 + 1.111111111 Combine like terms: 1.653652896 + 1.111111111 = 2.764764007 x = 2.764764007 Simplifying x = 2.764764007Subproblem 2
x + -1.111111111 = -1.653652896 Simplifying x + -1.111111111 = -1.653652896 Reorder the terms: -1.111111111 + x = -1.653652896 Solving -1.111111111 + x = -1.653652896 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.111111111' to each side of the equation. -1.111111111 + 1.111111111 + x = -1.653652896 + 1.111111111 Combine like terms: -1.111111111 + 1.111111111 = 0.000000000 0.000000000 + x = -1.653652896 + 1.111111111 x = -1.653652896 + 1.111111111 Combine like terms: -1.653652896 + 1.111111111 = -0.542541785 x = -0.542541785 Simplifying x = -0.542541785Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.764764007, -0.542541785}
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