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Simplifying x(2x + 1) = 18 Reorder the terms: x(1 + 2x) = 18 (1 * x + 2x * x) = 18 (1x + 2x2) = 18 Solving 1x + 2x2 = 18 Solving for variable 'x'. Reorder the terms: -18 + 1x + 2x2 = 18 + -18 Combine like terms: 18 + -18 = 0 -18 + 1x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -9 + 0.5x + x2 = 0 Move the constant term to the right: Add '9' to each side of the equation. -9 + 0.5x + 9 + x2 = 0 + 9 Reorder the terms: -9 + 9 + 0.5x + x2 = 0 + 9 Combine like terms: -9 + 9 = 0 0 + 0.5x + x2 = 0 + 9 0.5x + x2 = 0 + 9 Combine like terms: 0 + 9 = 9 0.5x + x2 = 9 The x term is 0.5x. Take half its coefficient (0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. 0.5x + 0.0625 + x2 = 9 + 0.0625 Reorder the terms: 0.0625 + 0.5x + x2 = 9 + 0.0625 Combine like terms: 9 + 0.0625 = 9.0625 0.0625 + 0.5x + x2 = 9.0625 Factor a perfect square on the left side: (x + 0.25)(x + 0.25) = 9.0625 Calculate the square root of the right side: 3.010398645 Break this problem into two subproblems by setting (x + 0.25) equal to 3.010398645 and -3.010398645.Subproblem 1
x + 0.25 = 3.010398645 Simplifying x + 0.25 = 3.010398645 Reorder the terms: 0.25 + x = 3.010398645 Solving 0.25 + x = 3.010398645 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = 3.010398645 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = 3.010398645 + -0.25 x = 3.010398645 + -0.25 Combine like terms: 3.010398645 + -0.25 = 2.760398645 x = 2.760398645 Simplifying x = 2.760398645Subproblem 2
x + 0.25 = -3.010398645 Simplifying x + 0.25 = -3.010398645 Reorder the terms: 0.25 + x = -3.010398645 Solving 0.25 + x = -3.010398645 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = -3.010398645 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = -3.010398645 + -0.25 x = -3.010398645 + -0.25 Combine like terms: -3.010398645 + -0.25 = -3.260398645 x = -3.260398645 Simplifying x = -3.260398645Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.760398645, -3.260398645}
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