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Simplifying 12w2 + 8w = -7w + -3w + 14 Reorder the terms: 8w + 12w2 = -7w + -3w + 14 Reorder the terms: 8w + 12w2 = 14 + -7w + -3w Combine like terms: -7w + -3w = -10w 8w + 12w2 = 14 + -10w Solving 8w + 12w2 = 14 + -10w Solving for variable 'w'. Reorder the terms: -14 + 8w + 10w + 12w2 = 14 + -10w + -14 + 10w Combine like terms: 8w + 10w = 18w -14 + 18w + 12w2 = 14 + -10w + -14 + 10w Reorder the terms: -14 + 18w + 12w2 = 14 + -14 + -10w + 10w Combine like terms: 14 + -14 = 0 -14 + 18w + 12w2 = 0 + -10w + 10w -14 + 18w + 12w2 = -10w + 10w Combine like terms: -10w + 10w = 0 -14 + 18w + 12w2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-7 + 9w + 6w2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-7 + 9w + 6w2)' equal to zero and attempt to solve: Simplifying -7 + 9w + 6w2 = 0 Solving -7 + 9w + 6w2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -1.166666667 + 1.5w + w2 = 0 Move the constant term to the right: Add '1.166666667' to each side of the equation. -1.166666667 + 1.5w + 1.166666667 + w2 = 0 + 1.166666667 Reorder the terms: -1.166666667 + 1.166666667 + 1.5w + w2 = 0 + 1.166666667 Combine like terms: -1.166666667 + 1.166666667 = 0.000000000 0.000000000 + 1.5w + w2 = 0 + 1.166666667 1.5w + w2 = 0 + 1.166666667 Combine like terms: 0 + 1.166666667 = 1.166666667 1.5w + w2 = 1.166666667 The w term is 1.5w. Take half its coefficient (0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. 1.5w + 0.5625 + w2 = 1.166666667 + 0.5625 Reorder the terms: 0.5625 + 1.5w + w2 = 1.166666667 + 0.5625 Combine like terms: 1.166666667 + 0.5625 = 1.729166667 0.5625 + 1.5w + w2 = 1.729166667 Factor a perfect square on the left side: (w + 0.75)(w + 0.75) = 1.729166667 Calculate the square root of the right side: 1.31497782 Break this problem into two subproblems by setting (w + 0.75) equal to 1.31497782 and -1.31497782.Subproblem 1
w + 0.75 = 1.31497782 Simplifying w + 0.75 = 1.31497782 Reorder the terms: 0.75 + w = 1.31497782 Solving 0.75 + w = 1.31497782 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + w = 1.31497782 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + w = 1.31497782 + -0.75 w = 1.31497782 + -0.75 Combine like terms: 1.31497782 + -0.75 = 0.56497782 w = 0.56497782 Simplifying w = 0.56497782Subproblem 2
w + 0.75 = -1.31497782 Simplifying w + 0.75 = -1.31497782 Reorder the terms: 0.75 + w = -1.31497782 Solving 0.75 + w = -1.31497782 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + w = -1.31497782 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + w = -1.31497782 + -0.75 w = -1.31497782 + -0.75 Combine like terms: -1.31497782 + -0.75 = -2.06497782 w = -2.06497782 Simplifying w = -2.06497782Solution
The solution to the problem is based on the solutions from the subproblems. w = {0.56497782, -2.06497782}Solution
w = {0.56497782, -2.06497782}
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