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Simplifying 2w(2w) + -3w = 135 Remove parenthesis around (2w) 2w * 2w + -3w = 135 Reorder the terms for easier multiplication: 2 * 2w * w + -3w = 135 Multiply 2 * 2 4w * w + -3w = 135 Multiply w * w 4w2 + -3w = 135 Reorder the terms: -3w + 4w2 = 135 Solving -3w + 4w2 = 135 Solving for variable 'w'. Reorder the terms: -135 + -3w + 4w2 = 135 + -135 Combine like terms: 135 + -135 = 0 -135 + -3w + 4w2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -33.75 + -0.75w + w2 = 0 Move the constant term to the right: Add '33.75' to each side of the equation. -33.75 + -0.75w + 33.75 + w2 = 0 + 33.75 Reorder the terms: -33.75 + 33.75 + -0.75w + w2 = 0 + 33.75 Combine like terms: -33.75 + 33.75 = 0.00 0.00 + -0.75w + w2 = 0 + 33.75 -0.75w + w2 = 0 + 33.75 Combine like terms: 0 + 33.75 = 33.75 -0.75w + w2 = 33.75 The w term is -0.75w. Take half its coefficient (-0.375). Square it (0.140625) and add it to both sides. Add '0.140625' to each side of the equation. -0.75w + 0.140625 + w2 = 33.75 + 0.140625 Reorder the terms: 0.140625 + -0.75w + w2 = 33.75 + 0.140625 Combine like terms: 33.75 + 0.140625 = 33.890625 0.140625 + -0.75w + w2 = 33.890625 Factor a perfect square on the left side: (w + -0.375)(w + -0.375) = 33.890625 Calculate the square root of the right side: 5.821565511 Break this problem into two subproblems by setting (w + -0.375) equal to 5.821565511 and -5.821565511.Subproblem 1
w + -0.375 = 5.821565511 Simplifying w + -0.375 = 5.821565511 Reorder the terms: -0.375 + w = 5.821565511 Solving -0.375 + w = 5.821565511 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '0.375' to each side of the equation. -0.375 + 0.375 + w = 5.821565511 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + w = 5.821565511 + 0.375 w = 5.821565511 + 0.375 Combine like terms: 5.821565511 + 0.375 = 6.196565511 w = 6.196565511 Simplifying w = 6.196565511Subproblem 2
w + -0.375 = -5.821565511 Simplifying w + -0.375 = -5.821565511 Reorder the terms: -0.375 + w = -5.821565511 Solving -0.375 + w = -5.821565511 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '0.375' to each side of the equation. -0.375 + 0.375 + w = -5.821565511 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + w = -5.821565511 + 0.375 w = -5.821565511 + 0.375 Combine like terms: -5.821565511 + 0.375 = -5.446565511 w = -5.446565511 Simplifying w = -5.446565511Solution
The solution to the problem is based on the solutions from the subproblems. w = {6.196565511, -5.446565511}
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