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Simplifying 31.5 = 2w2 + -4w Reorder the terms: 31.5 = -4w + 2w2 Solving 31.5 = -4w + 2w2 Solving for variable 'w'. Reorder the terms: 31.5 + 4w + -2w2 = -4w + 4w + 2w2 + -2w2 Combine like terms: -4w + 4w = 0 31.5 + 4w + -2w2 = 0 + 2w2 + -2w2 31.5 + 4w + -2w2 = 2w2 + -2w2 Combine like terms: 2w2 + -2w2 = 0 31.5 + 4w + -2w2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -15.75 + -2w + w2 = 0 Move the constant term to the right: Add '15.75' to each side of the equation. -15.75 + -2w + 15.75 + w2 = 0 + 15.75 Reorder the terms: -15.75 + 15.75 + -2w + w2 = 0 + 15.75 Combine like terms: -15.75 + 15.75 = 0.00 0.00 + -2w + w2 = 0 + 15.75 -2w + w2 = 0 + 15.75 Combine like terms: 0 + 15.75 = 15.75 -2w + w2 = 15.75 The w term is -2w. Take half its coefficient (-1). Square it (1) and add it to both sides. Add '1' to each side of the equation. -2w + 1 + w2 = 15.75 + 1 Reorder the terms: 1 + -2w + w2 = 15.75 + 1 Combine like terms: 15.75 + 1 = 16.75 1 + -2w + w2 = 16.75 Factor a perfect square on the left side: (w + -1)(w + -1) = 16.75 Calculate the square root of the right side: 4.092676386 Break this problem into two subproblems by setting (w + -1) equal to 4.092676386 and -4.092676386.Subproblem 1
w + -1 = 4.092676386 Simplifying w + -1 = 4.092676386 Reorder the terms: -1 + w = 4.092676386 Solving -1 + w = 4.092676386 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + w = 4.092676386 + 1 Combine like terms: -1 + 1 = 0 0 + w = 4.092676386 + 1 w = 4.092676386 + 1 Combine like terms: 4.092676386 + 1 = 5.092676386 w = 5.092676386 Simplifying w = 5.092676386Subproblem 2
w + -1 = -4.092676386 Simplifying w + -1 = -4.092676386 Reorder the terms: -1 + w = -4.092676386 Solving -1 + w = -4.092676386 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + w = -4.092676386 + 1 Combine like terms: -1 + 1 = 0 0 + w = -4.092676386 + 1 w = -4.092676386 + 1 Combine like terms: -4.092676386 + 1 = -3.092676386 w = -3.092676386 Simplifying w = -3.092676386Solution
The solution to the problem is based on the solutions from the subproblems. w = {5.092676386, -3.092676386}
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