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Simplifying 3w2 + -0.75w = 0.5 Reorder the terms: -0.75w + 3w2 = 0.5 Solving -0.75w + 3w2 = 0.5 Solving for variable 'w'. Reorder the terms: -0.5 + -0.75w + 3w2 = 0.5 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 -0.5 + -0.75w + 3w2 = 0.0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -0.1666666667 + -0.25w + w2 = 0.0 Move the constant term to the right: Add '0.1666666667' to each side of the equation. -0.1666666667 + -0.25w + 0.1666666667 + w2 = 0.0 + 0.1666666667 Reorder the terms: -0.1666666667 + 0.1666666667 + -0.25w + w2 = 0.0 + 0.1666666667 Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000 0.0000000000 + -0.25w + w2 = 0.0 + 0.1666666667 -0.25w + w2 = 0.0 + 0.1666666667 Combine like terms: 0.0 + 0.1666666667 = 0.1666666667 -0.25w + w2 = 0.1666666667 The w term is -0.25w. Take half its coefficient (-0.125). Square it (0.015625) and add it to both sides. Add '0.015625' to each side of the equation. -0.25w + 0.015625 + w2 = 0.1666666667 + 0.015625 Reorder the terms: 0.015625 + -0.25w + w2 = 0.1666666667 + 0.015625 Combine like terms: 0.1666666667 + 0.015625 = 0.1822916667 0.015625 + -0.25w + w2 = 0.1822916667 Factor a perfect square on the left side: (w + -0.125)(w + -0.125) = 0.1822916667 Calculate the square root of the right side: 0.426956282 Break this problem into two subproblems by setting (w + -0.125) equal to 0.426956282 and -0.426956282.Subproblem 1
w + -0.125 = 0.426956282 Simplifying w + -0.125 = 0.426956282 Reorder the terms: -0.125 + w = 0.426956282 Solving -0.125 + w = 0.426956282 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '0.125' to each side of the equation. -0.125 + 0.125 + w = 0.426956282 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + w = 0.426956282 + 0.125 w = 0.426956282 + 0.125 Combine like terms: 0.426956282 + 0.125 = 0.551956282 w = 0.551956282 Simplifying w = 0.551956282Subproblem 2
w + -0.125 = -0.426956282 Simplifying w + -0.125 = -0.426956282 Reorder the terms: -0.125 + w = -0.426956282 Solving -0.125 + w = -0.426956282 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '0.125' to each side of the equation. -0.125 + 0.125 + w = -0.426956282 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + w = -0.426956282 + 0.125 w = -0.426956282 + 0.125 Combine like terms: -0.426956282 + 0.125 = -0.301956282 w = -0.301956282 Simplifying w = -0.301956282Solution
The solution to the problem is based on the solutions from the subproblems. w = {0.551956282, -0.301956282}
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