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Simplifying (x(x + 1)) * 2 = 21 Reorder the terms: (x(1 + x)) * 2 = 21 ((1 * x + x * x)) * 2 = 21 ((1x + x2)) * 2 = 21 Reorder the terms for easier multiplication: 2(1x + x2) = 21 (1x * 2 + x2 * 2) = 21 (2x + 2x2) = 21 Solving 2x + 2x2 = 21 Solving for variable 'x'. Reorder the terms: -21 + 2x + 2x2 = 21 + -21 Combine like terms: 21 + -21 = 0 -21 + 2x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -10.5 + x + x2 = 0 Move the constant term to the right: Add '10.5' to each side of the equation. -10.5 + x + 10.5 + x2 = 0 + 10.5 Reorder the terms: -10.5 + 10.5 + x + x2 = 0 + 10.5 Combine like terms: -10.5 + 10.5 = 0.0 0.0 + x + x2 = 0 + 10.5 x + x2 = 0 + 10.5 Combine like terms: 0 + 10.5 = 10.5 x + x2 = 10.5 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. + 0.25 + x2 = 10.5 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + x2 = 10.5 + 0.25 Combine like terms: 10.5 + 0.25 = 10.75 1.25 + x2 = 10.75 Factor a perfect square on the left side: (x + 0.5)(x + 0.5) = 10.75 Calculate the square root of the right side: 3.278719262 Break this problem into two subproblems by setting (x + 0.5) equal to 3.278719262 and -3.278719262.Subproblem 1
x + 0.5 = 3.278719262 Simplifying x + 0.5 = 3.278719262 Reorder the terms: 0.5 + x = 3.278719262 Solving 0.5 + x = 3.278719262 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 = 3.278719262 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 3.278719262 + -0.5 x = 3.278719262 + -0.5 Combine like terms: 3.278719262 + -0.5 = 2.778719262 x = 2.778719262 Simplifying x = 2.778719262Subproblem 2
x + 0.5 = -3.278719262 Simplifying x + 0.5 = -3.278719262 Reorder the terms: 0.5 + x = -3.278719262 Solving 0.5 + x = -3.278719262 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 = -3.278719262 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -3.278719262 + -0.5 x = -3.278719262 + -0.5 Combine like terms: -3.278719262 + -0.5 = -3.778719262 x = -3.778719262 Simplifying x = -3.778719262Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.778719262, -3.778719262}
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