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Simplifying 11y2 + -42y + 13 = 0 Reorder the terms: 13 + -42y + 11y2 = 0 Solving 13 + -42y + 11y2 = 0 Solving for variable 'y'. Begin completing the square. Divide all terms by 11 the coefficient of the squared term: Divide each side by '11'. 1.181818182 + -3.818181818y + y2 = 0 Move the constant term to the right: Add '-1.181818182' to each side of the equation. 1.181818182 + -3.818181818y + -1.181818182 + y2 = 0 + -1.181818182 Reorder the terms: 1.181818182 + -1.181818182 + -3.818181818y + y2 = 0 + -1.181818182 Combine like terms: 1.181818182 + -1.181818182 = 0.000000000 0.000000000 + -3.818181818y + y2 = 0 + -1.181818182 -3.818181818y + y2 = 0 + -1.181818182 Combine like terms: 0 + -1.181818182 = -1.181818182 -3.818181818y + y2 = -1.181818182 The y term is -3.818181818y. Take half its coefficient (-1.909090909). Square it (3.644628099) and add it to both sides. Add '3.644628099' to each side of the equation. -3.818181818y + 3.644628099 + y2 = -1.181818182 + 3.644628099 Reorder the terms: 3.644628099 + -3.818181818y + y2 = -1.181818182 + 3.644628099 Combine like terms: -1.181818182 + 3.644628099 = 2.462809917 3.644628099 + -3.818181818y + y2 = 2.462809917 Factor a perfect square on the left side: (y + -1.909090909)(y + -1.909090909) = 2.462809917 Calculate the square root of the right side: 1.569334227 Break this problem into two subproblems by setting (y + -1.909090909) equal to 1.569334227 and -1.569334227.Subproblem 1
y + -1.909090909 = 1.569334227 Simplifying y + -1.909090909 = 1.569334227 Reorder the terms: -1.909090909 + y = 1.569334227 Solving -1.909090909 + y = 1.569334227 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.909090909' to each side of the equation. -1.909090909 + 1.909090909 + y = 1.569334227 + 1.909090909 Combine like terms: -1.909090909 + 1.909090909 = 0.000000000 0.000000000 + y = 1.569334227 + 1.909090909 y = 1.569334227 + 1.909090909 Combine like terms: 1.569334227 + 1.909090909 = 3.478425136 y = 3.478425136 Simplifying y = 3.478425136Subproblem 2
y + -1.909090909 = -1.569334227 Simplifying y + -1.909090909 = -1.569334227 Reorder the terms: -1.909090909 + y = -1.569334227 Solving -1.909090909 + y = -1.569334227 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.909090909' to each side of the equation. -1.909090909 + 1.909090909 + y = -1.569334227 + 1.909090909 Combine like terms: -1.909090909 + 1.909090909 = 0.000000000 0.000000000 + y = -1.569334227 + 1.909090909 y = -1.569334227 + 1.909090909 Combine like terms: -1.569334227 + 1.909090909 = 0.339756682 y = 0.339756682 Simplifying y = 0.339756682Solution
The solution to the problem is based on the solutions from the subproblems. y = {3.478425136, 0.339756682}
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