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Simplifying 0 = 3w2 + 20w + -45 Reorder the terms: 0 = -45 + 20w + 3w2 Solving 0 = -45 + 20w + 3w2 Solving for variable 'w'. Combine like terms: 0 + 45 = 45 45 + -20w + -3w2 = -45 + 20w + 3w2 + 45 + -20w + -3w2 Reorder the terms: 45 + -20w + -3w2 = -45 + 45 + 20w + -20w + 3w2 + -3w2 Combine like terms: -45 + 45 = 0 45 + -20w + -3w2 = 0 + 20w + -20w + 3w2 + -3w2 45 + -20w + -3w2 = 20w + -20w + 3w2 + -3w2 Combine like terms: 20w + -20w = 0 45 + -20w + -3w2 = 0 + 3w2 + -3w2 45 + -20w + -3w2 = 3w2 + -3w2 Combine like terms: 3w2 + -3w2 = 0 45 + -20w + -3w2 = 0 Begin completing the square. Divide all terms by -3 the coefficient of the squared term: Divide each side by '-3'. -15 + 6.666666667w + w2 = 0 Move the constant term to the right: Add '15' to each side of the equation. -15 + 6.666666667w + 15 + w2 = 0 + 15 Reorder the terms: -15 + 15 + 6.666666667w + w2 = 0 + 15 Combine like terms: -15 + 15 = 0 0 + 6.666666667w + w2 = 0 + 15 6.666666667w + w2 = 0 + 15 Combine like terms: 0 + 15 = 15 6.666666667w + w2 = 15 The w term is 6.666666667w. Take half its coefficient (3.333333334). Square it (11.11111112) and add it to both sides. Add '11.11111112' to each side of the equation. 6.666666667w + 11.11111112 + w2 = 15 + 11.11111112 Reorder the terms: 11.11111112 + 6.666666667w + w2 = 15 + 11.11111112 Combine like terms: 15 + 11.11111112 = 26.11111112 11.11111112 + 6.666666667w + w2 = 26.11111112 Factor a perfect square on the left side: (w + 3.333333334)(w + 3.333333334) = 26.11111112 Calculate the square root of the right side: 5.10990324 Break this problem into two subproblems by setting (w + 3.333333334) equal to 5.10990324 and -5.10990324.Subproblem 1
w + 3.333333334 = 5.10990324 Simplifying w + 3.333333334 = 5.10990324 Reorder the terms: 3.333333334 + w = 5.10990324 Solving 3.333333334 + w = 5.10990324 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-3.333333334' to each side of the equation. 3.333333334 + -3.333333334 + w = 5.10990324 + -3.333333334 Combine like terms: 3.333333334 + -3.333333334 = 0.000000000 0.000000000 + w = 5.10990324 + -3.333333334 w = 5.10990324 + -3.333333334 Combine like terms: 5.10990324 + -3.333333334 = 1.776569906 w = 1.776569906 Simplifying w = 1.776569906Subproblem 2
w + 3.333333334 = -5.10990324 Simplifying w + 3.333333334 = -5.10990324 Reorder the terms: 3.333333334 + w = -5.10990324 Solving 3.333333334 + w = -5.10990324 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-3.333333334' to each side of the equation. 3.333333334 + -3.333333334 + w = -5.10990324 + -3.333333334 Combine like terms: 3.333333334 + -3.333333334 = 0.000000000 0.000000000 + w = -5.10990324 + -3.333333334 w = -5.10990324 + -3.333333334 Combine like terms: -5.10990324 + -3.333333334 = -8.443236574 w = -8.443236574 Simplifying w = -8.443236574Solution
The solution to the problem is based on the solutions from the subproblems. w = {1.776569906, -8.443236574}
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