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Simplifying 81 = x(2x + -1) Reorder the terms: 81 = x(-1 + 2x) 81 = (-1 * x + 2x * x) 81 = (-1x + 2x2) Solving 81 = -1x + 2x2 Solving for variable 'x'. Reorder the terms: 81 + x + -2x2 = -1x + x + 2x2 + -2x2 Combine like terms: -1x + x = 0 81 + x + -2x2 = 0 + 2x2 + -2x2 81 + x + -2x2 = 2x2 + -2x2 Combine like terms: 2x2 + -2x2 = 0 81 + x + -2x2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -40.5 + -0.5x + x2 = 0 Move the constant term to the right: Add '40.5' to each side of the equation. -40.5 + -0.5x + 40.5 + x2 = 0 + 40.5 Reorder the terms: -40.5 + 40.5 + -0.5x + x2 = 0 + 40.5 Combine like terms: -40.5 + 40.5 = 0.0 0.0 + -0.5x + x2 = 0 + 40.5 -0.5x + x2 = 0 + 40.5 Combine like terms: 0 + 40.5 = 40.5 -0.5x + x2 = 40.5 The x term is x. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -0.5x + 0.25 + x2 = 40.5 + 0.25 Reorder the terms: 0.25 + -0.5x + x2 = 40.5 + 0.25 Combine like terms: 40.5 + 0.25 = 40.75 0.25 + -0.5x + x2 = 40.75 Factor a perfect square on the left side: (x + 0.5)(x + 0.5) = 40.75 Calculate the square root of the right side: 6.383572667 Break this problem into two subproblems by setting (x + 0.5) equal to 6.383572667 and -6.383572667.Subproblem 1
x + 0.5 = 6.383572667 Simplifying x + 0.5 = 6.383572667 Reorder the terms: 0.5 + x = 6.383572667 Solving 0.5 + x = 6.383572667 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = 6.383572667 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 6.383572667 + -0.5 x = 6.383572667 + -0.5 Combine like terms: 6.383572667 + -0.5 = 5.883572667 x = 5.883572667 Simplifying x = 5.883572667Subproblem 2
x + 0.5 = -6.383572667 Simplifying x + 0.5 = -6.383572667 Reorder the terms: 0.5 + x = -6.383572667 Solving 0.5 + x = -6.383572667 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = -6.383572667 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -6.383572667 + -0.5 x = -6.383572667 + -0.5 Combine like terms: -6.383572667 + -0.5 = -6.883572667 x = -6.883572667 Simplifying x = -6.883572667Solution
The solution to the problem is based on the solutions from the subproblems. x = {5.883572667, -6.883572667}
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