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Simplifying 14y2 + -41y + 21 = 0 Reorder the terms: 21 + -41y + 14y2 = 0 Solving 21 + -41y + 14y2 = 0 Solving for variable 'y'. Begin completing the square. Divide all terms by 14 the coefficient of the squared term: Divide each side by '14'. 1.5 + -2.928571429y + y2 = 0 Move the constant term to the right: Add '-1.5' to each side of the equation. 1.5 + -2.928571429y + -1.5 + y2 = 0 + -1.5 Reorder the terms: 1.5 + -1.5 + -2.928571429y + y2 = 0 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + -2.928571429y + y2 = 0 + -1.5 -2.928571429y + y2 = 0 + -1.5 Combine like terms: 0 + -1.5 = -1.5 -2.928571429y + y2 = -1.5 The y term is -2.928571429y. Take half its coefficient (-1.464285715). Square it (2.144132655) and add it to both sides. Add '2.144132655' to each side of the equation. -2.928571429y + 2.144132655 + y2 = -1.5 + 2.144132655 Reorder the terms: 2.144132655 + -2.928571429y + y2 = -1.5 + 2.144132655 Combine like terms: -1.5 + 2.144132655 = 0.644132655 2.144132655 + -2.928571429y + y2 = 0.644132655 Factor a perfect square on the left side: (y + -1.464285715)(y + -1.464285715) = 0.644132655 Calculate the square root of the right side: 0.802578753 Break this problem into two subproblems by setting (y + -1.464285715) equal to 0.802578753 and -0.802578753.Subproblem 1
y + -1.464285715 = 0.802578753 Simplifying y + -1.464285715 = 0.802578753 Reorder the terms: -1.464285715 + y = 0.802578753 Solving -1.464285715 + y = 0.802578753 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.464285715' to each side of the equation. -1.464285715 + 1.464285715 + y = 0.802578753 + 1.464285715 Combine like terms: -1.464285715 + 1.464285715 = 0.000000000 0.000000000 + y = 0.802578753 + 1.464285715 y = 0.802578753 + 1.464285715 Combine like terms: 0.802578753 + 1.464285715 = 2.266864468 y = 2.266864468 Simplifying y = 2.266864468Subproblem 2
y + -1.464285715 = -0.802578753 Simplifying y + -1.464285715 = -0.802578753 Reorder the terms: -1.464285715 + y = -0.802578753 Solving -1.464285715 + y = -0.802578753 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.464285715' to each side of the equation. -1.464285715 + 1.464285715 + y = -0.802578753 + 1.464285715 Combine like terms: -1.464285715 + 1.464285715 = 0.000000000 0.000000000 + y = -0.802578753 + 1.464285715 y = -0.802578753 + 1.464285715 Combine like terms: -0.802578753 + 1.464285715 = 0.661706962 y = 0.661706962 Simplifying y = 0.661706962Solution
The solution to the problem is based on the solutions from the subproblems. y = {2.266864468, 0.661706962}
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