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Simplifying 4w2 + 12w + -9 = 81 Reorder the terms: -9 + 12w + 4w2 = 81 Solving -9 + 12w + 4w2 = 81 Solving for variable 'w'. Reorder the terms: -9 + -81 + 12w + 4w2 = 81 + -81 Combine like terms: -9 + -81 = -90 -90 + 12w + 4w2 = 81 + -81 Combine like terms: 81 + -81 = 0 -90 + 12w + 4w2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-45 + 6w + 2w2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-45 + 6w + 2w2)' equal to zero and attempt to solve: Simplifying -45 + 6w + 2w2 = 0 Solving -45 + 6w + 2w2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -22.5 + 3w + w2 = 0 Move the constant term to the right: Add '22.5' to each side of the equation. -22.5 + 3w + 22.5 + w2 = 0 + 22.5 Reorder the terms: -22.5 + 22.5 + 3w + w2 = 0 + 22.5 Combine like terms: -22.5 + 22.5 = 0.0 0.0 + 3w + w2 = 0 + 22.5 3w + w2 = 0 + 22.5 Combine like terms: 0 + 22.5 = 22.5 3w + w2 = 22.5 The w term is 3w. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3w + 2.25 + w2 = 22.5 + 2.25 Reorder the terms: 2.25 + 3w + w2 = 22.5 + 2.25 Combine like terms: 22.5 + 2.25 = 24.75 2.25 + 3w + w2 = 24.75 Factor a perfect square on the left side: (w + 1.5)(w + 1.5) = 24.75 Calculate the square root of the right side: 4.974937186 Break this problem into two subproblems by setting (w + 1.5) equal to 4.974937186 and -4.974937186.Subproblem 1
w + 1.5 = 4.974937186 Simplifying w + 1.5 = 4.974937186 Reorder the terms: 1.5 + w = 4.974937186 Solving 1.5 + w = 4.974937186 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + w = 4.974937186 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + w = 4.974937186 + -1.5 w = 4.974937186 + -1.5 Combine like terms: 4.974937186 + -1.5 = 3.474937186 w = 3.474937186 Simplifying w = 3.474937186Subproblem 2
w + 1.5 = -4.974937186 Simplifying w + 1.5 = -4.974937186 Reorder the terms: 1.5 + w = -4.974937186 Solving 1.5 + w = -4.974937186 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + w = -4.974937186 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + w = -4.974937186 + -1.5 w = -4.974937186 + -1.5 Combine like terms: -4.974937186 + -1.5 = -6.474937186 w = -6.474937186 Simplifying w = -6.474937186Solution
The solution to the problem is based on the solutions from the subproblems. w = {3.474937186, -6.474937186}Solution
w = {3.474937186, -6.474937186}
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