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Simplifying 5y2 + -7y + 6 + -14y2 + -7y + 7 = 0 Reorder the terms: 6 + 7 + -7y + -7y + 5y2 + -14y2 = 0 Combine like terms: 6 + 7 = 13 13 + -7y + -7y + 5y2 + -14y2 = 0 Combine like terms: -7y + -7y = -14y 13 + -14y + 5y2 + -14y2 = 0 Combine like terms: 5y2 + -14y2 = -9y2 13 + -14y + -9y2 = 0 Solving 13 + -14y + -9y2 = 0 Solving for variable 'y'. Begin completing the square. Divide all terms by -9 the coefficient of the squared term: Divide each side by '-9'. -1.444444444 + 1.555555556y + y2 = 0 Move the constant term to the right: Add '1.444444444' to each side of the equation. -1.444444444 + 1.555555556y + 1.444444444 + y2 = 0 + 1.444444444 Reorder the terms: -1.444444444 + 1.444444444 + 1.555555556y + y2 = 0 + 1.444444444 Combine like terms: -1.444444444 + 1.444444444 = 0.000000000 0.000000000 + 1.555555556y + y2 = 0 + 1.444444444 1.555555556y + y2 = 0 + 1.444444444 Combine like terms: 0 + 1.444444444 = 1.444444444 1.555555556y + y2 = 1.444444444 The y term is 1.555555556y. Take half its coefficient (0.777777778). Square it (0.6049382720) and add it to both sides. Add '0.6049382720' to each side of the equation. 1.555555556y + 0.6049382720 + y2 = 1.444444444 + 0.6049382720 Reorder the terms: 0.6049382720 + 1.555555556y + y2 = 1.444444444 + 0.6049382720 Combine like terms: 1.444444444 + 0.6049382720 = 2.049382716 0.6049382720 + 1.555555556y + y2 = 2.049382716 Factor a perfect square on the left side: (y + 0.777777778)(y + 0.777777778) = 2.049382716 Calculate the square root of the right side: 1.431566525 Break this problem into two subproblems by setting (y + 0.777777778) equal to 1.431566525 and -1.431566525.Subproblem 1
y + 0.777777778 = 1.431566525 Simplifying y + 0.777777778 = 1.431566525 Reorder the terms: 0.777777778 + y = 1.431566525 Solving 0.777777778 + y = 1.431566525 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.777777778' to each side of the equation. 0.777777778 + -0.777777778 + y = 1.431566525 + -0.777777778 Combine like terms: 0.777777778 + -0.777777778 = 0.000000000 0.000000000 + y = 1.431566525 + -0.777777778 y = 1.431566525 + -0.777777778 Combine like terms: 1.431566525 + -0.777777778 = 0.653788747 y = 0.653788747 Simplifying y = 0.653788747Subproblem 2
y + 0.777777778 = -1.431566525 Simplifying y + 0.777777778 = -1.431566525 Reorder the terms: 0.777777778 + y = -1.431566525 Solving 0.777777778 + y = -1.431566525 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.777777778' to each side of the equation. 0.777777778 + -0.777777778 + y = -1.431566525 + -0.777777778 Combine like terms: 0.777777778 + -0.777777778 = 0.000000000 0.000000000 + y = -1.431566525 + -0.777777778 y = -1.431566525 + -0.777777778 Combine like terms: -1.431566525 + -0.777777778 = -2.209344303 y = -2.209344303 Simplifying y = -2.209344303Solution
The solution to the problem is based on the solutions from the subproblems. y = {0.653788747, -2.209344303}
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