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Simplifying 0 = 6w2 + -20w + -12 Reorder the terms: 0 = -12 + -20w + 6w2 Solving 0 = -12 + -20w + 6w2 Solving for variable 'w'. Combine like terms: 0 + 12 = 12 12 + 20w + -6w2 = -12 + -20w + 6w2 + 12 + 20w + -6w2 Reorder the terms: 12 + 20w + -6w2 = -12 + 12 + -20w + 20w + 6w2 + -6w2 Combine like terms: -12 + 12 = 0 12 + 20w + -6w2 = 0 + -20w + 20w + 6w2 + -6w2 12 + 20w + -6w2 = -20w + 20w + 6w2 + -6w2 Combine like terms: -20w + 20w = 0 12 + 20w + -6w2 = 0 + 6w2 + -6w2 12 + 20w + -6w2 = 6w2 + -6w2 Combine like terms: 6w2 + -6w2 = 0 12 + 20w + -6w2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(6 + 10w + -3w2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(6 + 10w + -3w2)' equal to zero and attempt to solve: Simplifying 6 + 10w + -3w2 = 0 Solving 6 + 10w + -3w2 = 0 Begin completing the square. Divide all terms by -3 the coefficient of the squared term: Divide each side by '-3'. -2 + -3.333333333w + w2 = 0 Move the constant term to the right: Add '2' to each side of the equation. -2 + -3.333333333w + 2 + w2 = 0 + 2 Reorder the terms: -2 + 2 + -3.333333333w + w2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + -3.333333333w + w2 = 0 + 2 -3.333333333w + w2 = 0 + 2 Combine like terms: 0 + 2 = 2 -3.333333333w + w2 = 2 The w term is -3.333333333w. Take half its coefficient (-1.666666667). Square it (2.777777779) and add it to both sides. Add '2.777777779' to each side of the equation. -3.333333333w + 2.777777779 + w2 = 2 + 2.777777779 Reorder the terms: 2.777777779 + -3.333333333w + w2 = 2 + 2.777777779 Combine like terms: 2 + 2.777777779 = 4.777777779 2.777777779 + -3.333333333w + w2 = 4.777777779 Factor a perfect square on the left side: (w + -1.666666667)(w + -1.666666667) = 4.777777779 Calculate the square root of the right side: 2.185812842 Break this problem into two subproblems by setting (w + -1.666666667) equal to 2.185812842 and -2.185812842.Subproblem 1
w + -1.666666667 = 2.185812842 Simplifying w + -1.666666667 = 2.185812842 Reorder the terms: -1.666666667 + w = 2.185812842 Solving -1.666666667 + w = 2.185812842 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '1.666666667' to each side of the equation. -1.666666667 + 1.666666667 + w = 2.185812842 + 1.666666667 Combine like terms: -1.666666667 + 1.666666667 = 0.000000000 0.000000000 + w = 2.185812842 + 1.666666667 w = 2.185812842 + 1.666666667 Combine like terms: 2.185812842 + 1.666666667 = 3.852479509 w = 3.852479509 Simplifying w = 3.852479509Subproblem 2
w + -1.666666667 = -2.185812842 Simplifying w + -1.666666667 = -2.185812842 Reorder the terms: -1.666666667 + w = -2.185812842 Solving -1.666666667 + w = -2.185812842 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '1.666666667' to each side of the equation. -1.666666667 + 1.666666667 + w = -2.185812842 + 1.666666667 Combine like terms: -1.666666667 + 1.666666667 = 0.000000000 0.000000000 + w = -2.185812842 + 1.666666667 w = -2.185812842 + 1.666666667 Combine like terms: -2.185812842 + 1.666666667 = -0.519146175 w = -0.519146175 Simplifying w = -0.519146175Solution
The solution to the problem is based on the solutions from the subproblems. w = {3.852479509, -0.519146175}Solution
w = {3.852479509, -0.519146175}
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