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Simplifying 3w2 + w = 20 Reorder the terms: w + 3w2 = 20 Solving w + 3w2 = 20 Solving for variable 'w'. Reorder the terms: -20 + w + 3w2 = 20 + -20 Combine like terms: 20 + -20 = 0 -20 + w + 3w2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -6.666666667 + 0.3333333333w + w2 = 0 Move the constant term to the right: Add '6.666666667' to each side of the equation. -6.666666667 + 0.3333333333w + 6.666666667 + w2 = 0 + 6.666666667 Reorder the terms: -6.666666667 + 6.666666667 + 0.3333333333w + w2 = 0 + 6.666666667 Combine like terms: -6.666666667 + 6.666666667 = 0.000000000 0.000000000 + 0.3333333333w + w2 = 0 + 6.666666667 0.3333333333w + w2 = 0 + 6.666666667 Combine like terms: 0 + 6.666666667 = 6.666666667 0.3333333333w + w2 = 6.666666667 The w term is w. 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. 0.3333333333w + 0.25 + w2 = 6.666666667 + 0.25 Reorder the terms: 0.25 + 0.3333333333w + w2 = 6.666666667 + 0.25 Combine like terms: 6.666666667 + 0.25 = 6.916666667 0.25 + 0.3333333333w + w2 = 6.916666667 Factor a perfect square on the left side: (w + 0.5)(w + 0.5) = 6.916666667 Calculate the square root of the right side: 2.62995564 Break this problem into two subproblems by setting (w + 0.5) equal to 2.62995564 and -2.62995564.Subproblem 1
w + 0.5 = 2.62995564 Simplifying w + 0.5 = 2.62995564 Reorder the terms: 0.5 + w = 2.62995564 Solving 0.5 + w = 2.62995564 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 = 2.62995564 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + w = 2.62995564 + -0.5 w = 2.62995564 + -0.5 Combine like terms: 2.62995564 + -0.5 = 2.12995564 w = 2.12995564 Simplifying w = 2.12995564Subproblem 2
w + 0.5 = -2.62995564 Simplifying w + 0.5 = -2.62995564 Reorder the terms: 0.5 + w = -2.62995564 Solving 0.5 + w = -2.62995564 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 = -2.62995564 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + w = -2.62995564 + -0.5 w = -2.62995564 + -0.5 Combine like terms: -2.62995564 + -0.5 = -3.12995564 w = -3.12995564 Simplifying w = -3.12995564Solution
The solution to the problem is based on the solutions from the subproblems. w = {2.12995564, -3.12995564}
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