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Simplifying 2w2 + 12w = 82 Reorder the terms: 12w + 2w2 = 82 Solving 12w + 2w2 = 82 Solving for variable 'w'. Reorder the terms: -82 + 12w + 2w2 = 82 + -82 Combine like terms: 82 + -82 = 0 -82 + 12w + 2w2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-41 + 6w + w2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-41 + 6w + w2)' equal to zero and attempt to solve: Simplifying -41 + 6w + w2 = 0 Solving -41 + 6w + w2 = 0 Begin completing the square. Move the constant term to the right: Add '41' to each side of the equation. -41 + 6w + 41 + w2 = 0 + 41 Reorder the terms: -41 + 41 + 6w + w2 = 0 + 41 Combine like terms: -41 + 41 = 0 0 + 6w + w2 = 0 + 41 6w + w2 = 0 + 41 Combine like terms: 0 + 41 = 41 6w + w2 = 41 The w term is 6w. Take half its coefficient (3). Square it (9) and add it to both sides. Add '9' to each side of the equation. 6w + 9 + w2 = 41 + 9 Reorder the terms: 9 + 6w + w2 = 41 + 9 Combine like terms: 41 + 9 = 50 9 + 6w + w2 = 50 Factor a perfect square on the left side: (w + 3)(w + 3) = 50 Calculate the square root of the right side: 7.071067812 Break this problem into two subproblems by setting (w + 3) equal to 7.071067812 and -7.071067812.Subproblem 1
w + 3 = 7.071067812 Simplifying w + 3 = 7.071067812 Reorder the terms: 3 + w = 7.071067812 Solving 3 + w = 7.071067812 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + w = 7.071067812 + -3 Combine like terms: 3 + -3 = 0 0 + w = 7.071067812 + -3 w = 7.071067812 + -3 Combine like terms: 7.071067812 + -3 = 4.071067812 w = 4.071067812 Simplifying w = 4.071067812Subproblem 2
w + 3 = -7.071067812 Simplifying w + 3 = -7.071067812 Reorder the terms: 3 + w = -7.071067812 Solving 3 + w = -7.071067812 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + w = -7.071067812 + -3 Combine like terms: 3 + -3 = 0 0 + w = -7.071067812 + -3 w = -7.071067812 + -3 Combine like terms: -7.071067812 + -3 = -10.071067812 w = -10.071067812 Simplifying w = -10.071067812Solution
The solution to the problem is based on the solutions from the subproblems. w = {4.071067812, -10.071067812}Solution
w = {4.071067812, -10.071067812}
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