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Simplifying 11s2 + 11sw + -15w2 = 0 Reorder the terms: 11sw + 11s2 + -15w2 = 0 Solving 11sw + 11s2 + -15w2 = 0 Solving for variable 's'. Begin completing the square. Divide all terms by 11 the coefficient of the squared term: Divide each side by '11'. sw + s2 + -1.363636364w2 = 0 Move the constant term to the right: Add '1.363636364w2' to each side of the equation. sw + s2 + -1.363636364w2 + 1.363636364w2 = 0 + 1.363636364w2 Combine like terms: -1.363636364w2 + 1.363636364w2 = 0.000000000 sw + s2 + 0.000000000 = 0 + 1.363636364w2 sw + s2 = 0 + 1.363636364w2 Remove the zero: sw + s2 = 1.363636364w2 The s term is sw. Take half its coefficient (0.5w). Square it (0.25w2) and add it to both sides. Add '0.25w2' to each side of the equation. w2 + s2 + 0.25w2 = 1.363636364w2 + 0.25w2 Reorder the terms: s2 + w2 + 0.25w2 = 1.363636364w2 + 0.25w2 Combine like terms: w2 + 0.25w2 = 1.25w2 s2 + 1.25w2 = 1.363636364w2 + 0.25w2 Combine like terms: 1.363636364w2 + 0.25w2 = 1.613636364w2 s2 + 1.25w2 = 1.613636364w2 Factor a perfect square on the left side: (s + 0.5w)(s + 0.5w) = 1.613636364w2 Calculate the square root of the right side: 1.270289874w Break this problem into two subproblems by setting (s + 0.5w) equal to 1.270289874w and -1.270289874w.Subproblem 1
s + 0.5w = 1.270289874w Simplifying s + 0.5w = 1.270289874w Solving s + 0.5w = 1.270289874w Solving for variable 's'. Move all terms containing s to the left, all other terms to the right. Add '-0.5w' to each side of the equation. s + 0.5w + -0.5w = 1.270289874w + -0.5w Combine like terms: 0.5w + -0.5w = 0.0 s + 0.0 = 1.270289874w + -0.5w s = 1.270289874w + -0.5w Combine like terms: 1.270289874w + -0.5w = 0.770289874w s = 0.770289874w Simplifying s = 0.770289874wSubproblem 2
s + 0.5w = -1.270289874w Simplifying s + 0.5w = -1.270289874w Solving s + 0.5w = -1.270289874w Solving for variable 's'. Move all terms containing s to the left, all other terms to the right. Add '-0.5w' to each side of the equation. s + 0.5w + -0.5w = -1.270289874w + -0.5w Combine like terms: 0.5w + -0.5w = 0.0 s + 0.0 = -1.270289874w + -0.5w s = -1.270289874w + -0.5w Combine like terms: -1.270289874w + -0.5w = -1.770289874w s = -1.770289874w Simplifying s = -1.770289874wSolution
The solution to the problem is based on the solutions from the subproblems. s = {0.770289874w, -1.770289874w}
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