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Simplifying 21 = w(2w + -2) Reorder the terms: 21 = w(-2 + 2w) 21 = (-2 * w + 2w * w) 21 = (-2w + 2w2) Solving 21 = -2w + 2w2 Solving for variable 'w'. Reorder the terms: 21 + 2w + -2w2 = -2w + 2w + 2w2 + -2w2 Combine like terms: -2w + 2w = 0 21 + 2w + -2w2 = 0 + 2w2 + -2w2 21 + 2w + -2w2 = 2w2 + -2w2 Combine like terms: 2w2 + -2w2 = 0 21 + 2w + -2w2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -10.5 + -1w + w2 = 0 Move the constant term to the right: Add '10.5' to each side of the equation. -10.5 + -1w + 10.5 + w2 = 0 + 10.5 Reorder the terms: -10.5 + 10.5 + -1w + w2 = 0 + 10.5 Combine like terms: -10.5 + 10.5 = 0.0 0.0 + -1w + w2 = 0 + 10.5 -1w + w2 = 0 + 10.5 Combine like terms: 0 + 10.5 = 10.5 -1w + w2 = 10.5 The w term is -1w. 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. -1w + 0.25 + w2 = 10.5 + 0.25 Reorder the terms: 0.25 + -1w + w2 = 10.5 + 0.25 Combine like terms: 10.5 + 0.25 = 10.75 0.25 + -1w + w2 = 10.75 Factor a perfect square on the left side: (w + -0.5)(w + -0.5) = 10.75 Calculate the square root of the right side: 3.278719262 Break this problem into two subproblems by setting (w + -0.5) equal to 3.278719262 and -3.278719262.Subproblem 1
w + -0.5 = 3.278719262 Simplifying w + -0.5 = 3.278719262 Reorder the terms: -0.5 + w = 3.278719262 Solving -0.5 + w = 3.278719262 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + w = 3.278719262 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + w = 3.278719262 + 0.5 w = 3.278719262 + 0.5 Combine like terms: 3.278719262 + 0.5 = 3.778719262 w = 3.778719262 Simplifying w = 3.778719262Subproblem 2
w + -0.5 = -3.278719262 Simplifying w + -0.5 = -3.278719262 Reorder the terms: -0.5 + w = -3.278719262 Solving -0.5 + w = -3.278719262 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + w = -3.278719262 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + w = -3.278719262 + 0.5 w = -3.278719262 + 0.5 Combine like terms: -3.278719262 + 0.5 = -2.778719262 w = -2.778719262 Simplifying w = -2.778719262Solution
The solution to the problem is based on the solutions from the subproblems. w = {3.778719262, -2.778719262}
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