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Simplifying x(2x + 1) = 19 Reorder the terms: x(1 + 2x) = 19 (1 * x + 2x * x) = 19 (1x + 2x2) = 19 Solving 1x + 2x2 = 19 Solving for variable 'x'. Reorder the terms: -19 + 1x + 2x2 = 19 + -19 Combine like terms: 19 + -19 = 0 -19 + 1x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -9.5 + 0.5x + x2 = 0 Move the constant term to the right: Add '9.5' to each side of the equation. -9.5 + 0.5x + 9.5 + x2 = 0 + 9.5 Reorder the terms: -9.5 + 9.5 + 0.5x + x2 = 0 + 9.5 Combine like terms: -9.5 + 9.5 = 0.0 0.0 + 0.5x + x2 = 0 + 9.5 0.5x + x2 = 0 + 9.5 Combine like terms: 0 + 9.5 = 9.5 0.5x + x2 = 9.5 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 = 9.5 + 0.0625 Reorder the terms: 0.0625 + 0.5x + x2 = 9.5 + 0.0625 Combine like terms: 9.5 + 0.0625 = 9.5625 0.0625 + 0.5x + x2 = 9.5625 Factor a perfect square on the left side: (x + 0.25)(x + 0.25) = 9.5625 Calculate the square root of the right side: 3.092329219 Break this problem into two subproblems by setting (x + 0.25) equal to 3.092329219 and -3.092329219.Subproblem 1
x + 0.25 = 3.092329219 Simplifying x + 0.25 = 3.092329219 Reorder the terms: 0.25 + x = 3.092329219 Solving 0.25 + x = 3.092329219 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 = 3.092329219 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = 3.092329219 + -0.25 x = 3.092329219 + -0.25 Combine like terms: 3.092329219 + -0.25 = 2.842329219 x = 2.842329219 Simplifying x = 2.842329219Subproblem 2
x + 0.25 = -3.092329219 Simplifying x + 0.25 = -3.092329219 Reorder the terms: 0.25 + x = -3.092329219 Solving 0.25 + x = -3.092329219 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 = -3.092329219 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = -3.092329219 + -0.25 x = -3.092329219 + -0.25 Combine like terms: -3.092329219 + -0.25 = -3.342329219 x = -3.342329219 Simplifying x = -3.342329219Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.842329219, -3.342329219}
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