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Simplifying 5y2 + 3y2 + 25y + 15 = 0 Reorder the terms: 15 + 25y + 5y2 + 3y2 = 0 Combine like terms: 5y2 + 3y2 = 8y2 15 + 25y + 8y2 = 0 Solving 15 + 25y + 8y2 = 0 Solving for variable 'y'. Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. 1.875 + 3.125y + y2 = 0 Move the constant term to the right: Add '-1.875' to each side of the equation. 1.875 + 3.125y + -1.875 + y2 = 0 + -1.875 Reorder the terms: 1.875 + -1.875 + 3.125y + y2 = 0 + -1.875 Combine like terms: 1.875 + -1.875 = 0.000 0.000 + 3.125y + y2 = 0 + -1.875 3.125y + y2 = 0 + -1.875 Combine like terms: 0 + -1.875 = -1.875 3.125y + y2 = -1.875 The y term is 3.125y. Take half its coefficient (1.5625). Square it (2.44140625) and add it to both sides. Add '2.44140625' to each side of the equation. 3.125y + 2.44140625 + y2 = -1.875 + 2.44140625 Reorder the terms: 2.44140625 + 3.125y + y2 = -1.875 + 2.44140625 Combine like terms: -1.875 + 2.44140625 = 0.56640625 2.44140625 + 3.125y + y2 = 0.56640625 Factor a perfect square on the left side: (y + 1.5625)(y + 1.5625) = 0.56640625 Calculate the square root of the right side: 0.752599661 Break this problem into two subproblems by setting (y + 1.5625) equal to 0.752599661 and -0.752599661.Subproblem 1
y + 1.5625 = 0.752599661 Simplifying y + 1.5625 = 0.752599661 Reorder the terms: 1.5625 + y = 0.752599661 Solving 1.5625 + y = 0.752599661 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.5625' to each side of the equation. 1.5625 + -1.5625 + y = 0.752599661 + -1.5625 Combine like terms: 1.5625 + -1.5625 = 0.0000 0.0000 + y = 0.752599661 + -1.5625 y = 0.752599661 + -1.5625 Combine like terms: 0.752599661 + -1.5625 = -0.809900339 y = -0.809900339 Simplifying y = -0.809900339Subproblem 2
y + 1.5625 = -0.752599661 Simplifying y + 1.5625 = -0.752599661 Reorder the terms: 1.5625 + y = -0.752599661 Solving 1.5625 + y = -0.752599661 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.5625' to each side of the equation. 1.5625 + -1.5625 + y = -0.752599661 + -1.5625 Combine like terms: 1.5625 + -1.5625 = 0.0000 0.0000 + y = -0.752599661 + -1.5625 y = -0.752599661 + -1.5625 Combine like terms: -0.752599661 + -1.5625 = -2.315099661 y = -2.315099661 Simplifying y = -2.315099661Solution
The solution to the problem is based on the solutions from the subproblems. y = {-0.809900339, -2.315099661}
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