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Simplifying ((-0.75 + -1s)(-0.75 + -1s)) + -1(0.25 * 0.25) = 0 Multiply (-0.75 + -1s) * (-0.75 + -1s) ((-0.75(-0.75 + -1s) + -1s * (-0.75 + -1s))) + -1(0.25 * 0.25) = 0 (((-0.75 * -0.75 + -1s * -0.75) + -1s * (-0.75 + -1s))) + -1(0.25 * 0.25) = 0 (((0.5625 + 0.75s) + -1s * (-0.75 + -1s))) + -1(0.25 * 0.25) = 0 ((0.5625 + 0.75s + (-0.75 * -1s + -1s * -1s))) + -1(0.25 * 0.25) = 0 ((0.5625 + 0.75s + (0.75s + 1s2))) + -1(0.25 * 0.25) = 0 Combine like terms: 0.75s + 0.75s = 1.5s ((0.5625 + 1.5s + 1s2)) + -1(0.25 * 0.25) = 0 (0.5625 + 1.5s + 1s2) + -1(0.25 * 0.25) = 0 Remove parenthesis around (0.5625 + 1.5s + 1s2) 0.5625 + 1.5s + 1s2 + -1(0.25 * 0.25) = 0 Multiply 0.25 * 0.25 0.5625 + 1.5s + 1s2 + -1(0.0625) = 0 Multiply -1 * 0.0625 0.5625 + 1.5s + 1s2 + -0.0625 = 0 Reorder the terms: 0.5625 + -0.0625 + 1.5s + 1s2 = 0 Combine like terms: 0.5625 + -0.0625 = 0.5 0.5 + 1.5s + 1s2 = 0 Solving 0.5 + 1.5s + 1s2 = 0 Solving for variable 's'. Begin completing the square. Move the constant term to the right: Add '-0.5' to each side of the equation. 0.5 + 1.5s + -0.5 + s2 = 0 + -0.5 Reorder the terms: 0.5 + -0.5 + 1.5s + s2 = 0 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + 1.5s + s2 = 0 + -0.5 1.5s + s2 = 0 + -0.5 Combine like terms: 0 + -0.5 = -0.5 1.5s + s2 = -0.5 The s term is 1.5s. Take half its coefficient (0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. 1.5s + 0.5625 + s2 = -0.5 + 0.5625 Reorder the terms: 0.5625 + 1.5s + s2 = -0.5 + 0.5625 Combine like terms: -0.5 + 0.5625 = 0.0625 0.5625 + 1.5s + s2 = 0.0625 Factor a perfect square on the left side: (s + 0.75)(s + 0.75) = 0.0625 Calculate the square root of the right side: 0.25 Break this problem into two subproblems by setting (s + 0.75) equal to 0.25 and -0.25.Subproblem 1
s + 0.75 = 0.25 Simplifying s + 0.75 = 0.25 Reorder the terms: 0.75 + s = 0.25 Solving 0.75 + s = 0.25 Solving for variable 's'. Move all terms containing s to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + s = 0.25 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + s = 0.25 + -0.75 s = 0.25 + -0.75 Combine like terms: 0.25 + -0.75 = -0.5 s = -0.5 Simplifying s = -0.5Subproblem 2
s + 0.75 = -0.25 Simplifying s + 0.75 = -0.25 Reorder the terms: 0.75 + s = -0.25 Solving 0.75 + s = -0.25 Solving for variable 's'. Move all terms containing s to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + s = -0.25 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + s = -0.25 + -0.75 s = -0.25 + -0.75 Combine like terms: -0.25 + -0.75 = -1 s = -1 Simplifying s = -1Solution
The solution to the problem is based on the solutions from the subproblems. s = {-0.5, -1}
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