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Simplifying 5y2 + 3y + 10y + -2 = 0 Reorder the terms: -2 + 3y + 10y + 5y2 = 0 Combine like terms: 3y + 10y = 13y -2 + 13y + 5y2 = 0 Solving -2 + 13y + 5y2 = 0 Solving for variable 'y'. Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -0.4 + 2.6y + y2 = 0 Move the constant term to the right: Add '0.4' to each side of the equation. -0.4 + 2.6y + 0.4 + y2 = 0 + 0.4 Reorder the terms: -0.4 + 0.4 + 2.6y + y2 = 0 + 0.4 Combine like terms: -0.4 + 0.4 = 0.0 0.0 + 2.6y + y2 = 0 + 0.4 2.6y + y2 = 0 + 0.4 Combine like terms: 0 + 0.4 = 0.4 2.6y + y2 = 0.4 The y term is 2.6y. Take half its coefficient (1.3). Square it (1.69) and add it to both sides. Add '1.69' to each side of the equation. 2.6y + 1.69 + y2 = 0.4 + 1.69 Reorder the terms: 1.69 + 2.6y + y2 = 0.4 + 1.69 Combine like terms: 0.4 + 1.69 = 2.09 1.69 + 2.6y + y2 = 2.09 Factor a perfect square on the left side: (y + 1.3)(y + 1.3) = 2.09 Calculate the square root of the right side: 1.445683229 Break this problem into two subproblems by setting (y + 1.3) equal to 1.445683229 and -1.445683229.Subproblem 1
y + 1.3 = 1.445683229 Simplifying y + 1.3 = 1.445683229 Reorder the terms: 1.3 + y = 1.445683229 Solving 1.3 + y = 1.445683229 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.3' to each side of the equation. 1.3 + -1.3 + y = 1.445683229 + -1.3 Combine like terms: 1.3 + -1.3 = 0.0 0.0 + y = 1.445683229 + -1.3 y = 1.445683229 + -1.3 Combine like terms: 1.445683229 + -1.3 = 0.145683229 y = 0.145683229 Simplifying y = 0.145683229Subproblem 2
y + 1.3 = -1.445683229 Simplifying y + 1.3 = -1.445683229 Reorder the terms: 1.3 + y = -1.445683229 Solving 1.3 + y = -1.445683229 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.3' to each side of the equation. 1.3 + -1.3 + y = -1.445683229 + -1.3 Combine like terms: 1.3 + -1.3 = 0.0 0.0 + y = -1.445683229 + -1.3 y = -1.445683229 + -1.3 Combine like terms: -1.445683229 + -1.3 = -2.745683229 y = -2.745683229 Simplifying y = -2.745683229Solution
The solution to the problem is based on the solutions from the subproblems. y = {0.145683229, -2.745683229}
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