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Simplifying 90w2 + -30w + -25 = 0 Reorder the terms: -25 + -30w + 90w2 = 0 Solving -25 + -30w + 90w2 = 0 Solving for variable 'w'. Factor out the Greatest Common Factor (GCF), '5'. 5(-5 + -6w + 18w2) = 0 Ignore the factor 5.Subproblem 1
Set the factor '(-5 + -6w + 18w2)' equal to zero and attempt to solve: Simplifying -5 + -6w + 18w2 = 0 Solving -5 + -6w + 18w2 = 0 Begin completing the square. Divide all terms by 18 the coefficient of the squared term: Divide each side by '18'. -0.2777777778 + -0.3333333333w + w2 = 0 Move the constant term to the right: Add '0.2777777778' to each side of the equation. -0.2777777778 + -0.3333333333w + 0.2777777778 + w2 = 0 + 0.2777777778 Reorder the terms: -0.2777777778 + 0.2777777778 + -0.3333333333w + w2 = 0 + 0.2777777778 Combine like terms: -0.2777777778 + 0.2777777778 = 0.0000000000 0.0000000000 + -0.3333333333w + w2 = 0 + 0.2777777778 -0.3333333333w + w2 = 0 + 0.2777777778 Combine like terms: 0 + 0.2777777778 = 0.2777777778 -0.3333333333w + w2 = 0.2777777778 The w term is -0.3333333333w. Take half its coefficient (-0.1666666667). Square it (0.02777777779) and add it to both sides. Add '0.02777777779' to each side of the equation. -0.3333333333w + 0.02777777779 + w2 = 0.2777777778 + 0.02777777779 Reorder the terms: 0.02777777779 + -0.3333333333w + w2 = 0.2777777778 + 0.02777777779 Combine like terms: 0.2777777778 + 0.02777777779 = 0.30555555559 0.02777777779 + -0.3333333333w + w2 = 0.30555555559 Factor a perfect square on the left side: (w + -0.1666666667)(w + -0.1666666667) = 0.30555555559 Calculate the square root of the right side: 0.552770798 Break this problem into two subproblems by setting (w + -0.1666666667) equal to 0.552770798 and -0.552770798.Subproblem 1
w + -0.1666666667 = 0.552770798 Simplifying w + -0.1666666667 = 0.552770798 Reorder the terms: -0.1666666667 + w = 0.552770798 Solving -0.1666666667 + w = 0.552770798 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '0.1666666667' to each side of the equation. -0.1666666667 + 0.1666666667 + w = 0.552770798 + 0.1666666667 Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000 0.0000000000 + w = 0.552770798 + 0.1666666667 w = 0.552770798 + 0.1666666667 Combine like terms: 0.552770798 + 0.1666666667 = 0.7194374647 w = 0.7194374647 Simplifying w = 0.7194374647Subproblem 2
w + -0.1666666667 = -0.552770798 Simplifying w + -0.1666666667 = -0.552770798 Reorder the terms: -0.1666666667 + w = -0.552770798 Solving -0.1666666667 + w = -0.552770798 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '0.1666666667' to each side of the equation. -0.1666666667 + 0.1666666667 + w = -0.552770798 + 0.1666666667 Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000 0.0000000000 + w = -0.552770798 + 0.1666666667 w = -0.552770798 + 0.1666666667 Combine like terms: -0.552770798 + 0.1666666667 = -0.3861041313 w = -0.3861041313 Simplifying w = -0.3861041313Solution
The solution to the problem is based on the solutions from the subproblems. w = {0.7194374647, -0.3861041313}Solution
w = {0.7194374647, -0.3861041313}
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