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Simplifying -9w2 + 2w + 6 = 0 Reorder the terms: 6 + 2w + -9w2 = 0 Solving 6 + 2w + -9w2 = 0 Solving for variable 'w'. Begin completing the square. Divide all terms by -9 the coefficient of the squared term: Divide each side by '-9'. -0.6666666667 + -0.2222222222w + w2 = 0 Move the constant term to the right: Add '0.6666666667' to each side of the equation. -0.6666666667 + -0.2222222222w + 0.6666666667 + w2 = 0 + 0.6666666667 Reorder the terms: -0.6666666667 + 0.6666666667 + -0.2222222222w + w2 = 0 + 0.6666666667 Combine like terms: -0.6666666667 + 0.6666666667 = 0.0000000000 0.0000000000 + -0.2222222222w + w2 = 0 + 0.6666666667 -0.2222222222w + w2 = 0 + 0.6666666667 Combine like terms: 0 + 0.6666666667 = 0.6666666667 -0.2222222222w + w2 = 0.6666666667 The w term is -0.2222222222w. Take half its coefficient (-0.1111111111). Square it (0.01234567901) and add it to both sides. Add '0.01234567901' to each side of the equation. -0.2222222222w + 0.01234567901 + w2 = 0.6666666667 + 0.01234567901 Reorder the terms: 0.01234567901 + -0.2222222222w + w2 = 0.6666666667 + 0.01234567901 Combine like terms: 0.6666666667 + 0.01234567901 = 0.67901234571 0.01234567901 + -0.2222222222w + w2 = 0.67901234571 Factor a perfect square on the left side: (w + -0.1111111111)(w + -0.1111111111) = 0.67901234571 Calculate the square root of the right side: 0.824022054 Break this problem into two subproblems by setting (w + -0.1111111111) equal to 0.824022054 and -0.824022054.Subproblem 1
w + -0.1111111111 = 0.824022054 Simplifying w + -0.1111111111 = 0.824022054 Reorder the terms: -0.1111111111 + w = 0.824022054 Solving -0.1111111111 + w = 0.824022054 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '0.1111111111' to each side of the equation. -0.1111111111 + 0.1111111111 + w = 0.824022054 + 0.1111111111 Combine like terms: -0.1111111111 + 0.1111111111 = 0.0000000000 0.0000000000 + w = 0.824022054 + 0.1111111111 w = 0.824022054 + 0.1111111111 Combine like terms: 0.824022054 + 0.1111111111 = 0.9351331651 w = 0.9351331651 Simplifying w = 0.9351331651Subproblem 2
w + -0.1111111111 = -0.824022054 Simplifying w + -0.1111111111 = -0.824022054 Reorder the terms: -0.1111111111 + w = -0.824022054 Solving -0.1111111111 + w = -0.824022054 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '0.1111111111' to each side of the equation. -0.1111111111 + 0.1111111111 + w = -0.824022054 + 0.1111111111 Combine like terms: -0.1111111111 + 0.1111111111 = 0.0000000000 0.0000000000 + w = -0.824022054 + 0.1111111111 w = -0.824022054 + 0.1111111111 Combine like terms: -0.824022054 + 0.1111111111 = -0.7129109429 w = -0.7129109429 Simplifying w = -0.7129109429Solution
The solution to the problem is based on the solutions from the subproblems. w = {0.9351331651, -0.7129109429}
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