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Simplifying 24y2 + 105y + 105 = 0 Reorder the terms: 105 + 105y + 24y2 = 0 Solving 105 + 105y + 24y2 = 0 Solving for variable 'y'. Factor out the Greatest Common Factor (GCF), '3'. 3(35 + 35y + 8y2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(35 + 35y + 8y2)' equal to zero and attempt to solve: Simplifying 35 + 35y + 8y2 = 0 Solving 35 + 35y + 8y2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. 4.375 + 4.375y + y2 = 0 Move the constant term to the right: Add '-4.375' to each side of the equation. 4.375 + 4.375y + -4.375 + y2 = 0 + -4.375 Reorder the terms: 4.375 + -4.375 + 4.375y + y2 = 0 + -4.375 Combine like terms: 4.375 + -4.375 = 0.000 0.000 + 4.375y + y2 = 0 + -4.375 4.375y + y2 = 0 + -4.375 Combine like terms: 0 + -4.375 = -4.375 4.375y + y2 = -4.375 The y term is 4.375y. Take half its coefficient (2.1875). Square it (4.78515625) and add it to both sides. Add '4.78515625' to each side of the equation. 4.375y + 4.78515625 + y2 = -4.375 + 4.78515625 Reorder the terms: 4.78515625 + 4.375y + y2 = -4.375 + 4.78515625 Combine like terms: -4.375 + 4.78515625 = 0.41015625 4.78515625 + 4.375y + y2 = 0.41015625 Factor a perfect square on the left side: (y + 2.1875)(y + 2.1875) = 0.41015625 Calculate the square root of the right side: 0.640434423 Break this problem into two subproblems by setting (y + 2.1875) equal to 0.640434423 and -0.640434423.Subproblem 1
y + 2.1875 = 0.640434423 Simplifying y + 2.1875 = 0.640434423 Reorder the terms: 2.1875 + y = 0.640434423 Solving 2.1875 + y = 0.640434423 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-2.1875' to each side of the equation. 2.1875 + -2.1875 + y = 0.640434423 + -2.1875 Combine like terms: 2.1875 + -2.1875 = 0.0000 0.0000 + y = 0.640434423 + -2.1875 y = 0.640434423 + -2.1875 Combine like terms: 0.640434423 + -2.1875 = -1.547065577 y = -1.547065577 Simplifying y = -1.547065577Subproblem 2
y + 2.1875 = -0.640434423 Simplifying y + 2.1875 = -0.640434423 Reorder the terms: 2.1875 + y = -0.640434423 Solving 2.1875 + y = -0.640434423 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-2.1875' to each side of the equation. 2.1875 + -2.1875 + y = -0.640434423 + -2.1875 Combine like terms: 2.1875 + -2.1875 = 0.0000 0.0000 + y = -0.640434423 + -2.1875 y = -0.640434423 + -2.1875 Combine like terms: -0.640434423 + -2.1875 = -2.827934423 y = -2.827934423 Simplifying y = -2.827934423Solution
The solution to the problem is based on the solutions from the subproblems. y = {-1.547065577, -2.827934423}Solution
y = {-1.547065577, -2.827934423}
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