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Simplifying 14w4 + -10w3 + -70w2 = 0 Reorder the terms: -70w2 + -10w3 + 14w4 = 0 Solving -70w2 + -10w3 + 14w4 = 0 Solving for variable 'w'. Factor out the Greatest Common Factor (GCF), '2w2'. 2w2(-35 + -5w + 7w2) = 0 Ignore the factor 2.Subproblem 1
Set the factor 'w2' equal to zero and attempt to solve: Simplifying w2 = 0 Solving w2 = 0 Move all terms containing w to the left, all other terms to the right. Simplifying w2 = 0 Take the square root of each side: w = {0}Subproblem 2
Set the factor '(-35 + -5w + 7w2)' equal to zero and attempt to solve: Simplifying -35 + -5w + 7w2 = 0 Solving -35 + -5w + 7w2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -5 + -0.7142857143w + w2 = 0 Move the constant term to the right: Add '5' to each side of the equation. -5 + -0.7142857143w + 5 + w2 = 0 + 5 Reorder the terms: -5 + 5 + -0.7142857143w + w2 = 0 + 5 Combine like terms: -5 + 5 = 0 0 + -0.7142857143w + w2 = 0 + 5 -0.7142857143w + w2 = 0 + 5 Combine like terms: 0 + 5 = 5 -0.7142857143w + w2 = 5 The w term is -0.7142857143w. Take half its coefficient (-0.3571428572). Square it (0.1275510204) and add it to both sides. Add '0.1275510204' to each side of the equation. -0.7142857143w + 0.1275510204 + w2 = 5 + 0.1275510204 Reorder the terms: 0.1275510204 + -0.7142857143w + w2 = 5 + 0.1275510204 Combine like terms: 5 + 0.1275510204 = 5.1275510204 0.1275510204 + -0.7142857143w + w2 = 5.1275510204 Factor a perfect square on the left side: (w + -0.3571428572)(w + -0.3571428572) = 5.1275510204 Calculate the square root of the right side: 2.264409641 Break this problem into two subproblems by setting (w + -0.3571428572) equal to 2.264409641 and -2.264409641.Subproblem 1
w + -0.3571428572 = 2.264409641 Simplifying w + -0.3571428572 = 2.264409641 Reorder the terms: -0.3571428572 + w = 2.264409641 Solving -0.3571428572 + w = 2.264409641 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '0.3571428572' to each side of the equation. -0.3571428572 + 0.3571428572 + w = 2.264409641 + 0.3571428572 Combine like terms: -0.3571428572 + 0.3571428572 = 0.0000000000 0.0000000000 + w = 2.264409641 + 0.3571428572 w = 2.264409641 + 0.3571428572 Combine like terms: 2.264409641 + 0.3571428572 = 2.6215524982 w = 2.6215524982 Simplifying w = 2.6215524982Subproblem 2
w + -0.3571428572 = -2.264409641 Simplifying w + -0.3571428572 = -2.264409641 Reorder the terms: -0.3571428572 + w = -2.264409641 Solving -0.3571428572 + w = -2.264409641 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '0.3571428572' to each side of the equation. -0.3571428572 + 0.3571428572 + w = -2.264409641 + 0.3571428572 Combine like terms: -0.3571428572 + 0.3571428572 = 0.0000000000 0.0000000000 + w = -2.264409641 + 0.3571428572 w = -2.264409641 + 0.3571428572 Combine like terms: -2.264409641 + 0.3571428572 = -1.9072667838 w = -1.9072667838 Simplifying w = -1.9072667838Solution
The solution to the problem is based on the solutions from the subproblems. w = {2.6215524982, -1.9072667838}Solution
w = {0, 2.6215524982, -1.9072667838}
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