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Simplifying 9y2 + -6y = 1 Reorder the terms: -6y + 9y2 = 1 Solving -6y + 9y2 = 1 Solving for variable 'y'. Reorder the terms: -1 + -6y + 9y2 = 1 + -1 Combine like terms: 1 + -1 = 0 -1 + -6y + 9y2 = 0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -0.1111111111 + -0.6666666667y + y2 = 0 Move the constant term to the right: Add '0.1111111111' to each side of the equation. -0.1111111111 + -0.6666666667y + 0.1111111111 + y2 = 0 + 0.1111111111 Reorder the terms: -0.1111111111 + 0.1111111111 + -0.6666666667y + y2 = 0 + 0.1111111111 Combine like terms: -0.1111111111 + 0.1111111111 = 0.0000000000 0.0000000000 + -0.6666666667y + y2 = 0 + 0.1111111111 -0.6666666667y + y2 = 0 + 0.1111111111 Combine like terms: 0 + 0.1111111111 = 0.1111111111 -0.6666666667y + y2 = 0.1111111111 The y term is -0.6666666667y. Take half its coefficient (-0.3333333334). Square it (0.1111111112) and add it to both sides. Add '0.1111111112' to each side of the equation. -0.6666666667y + 0.1111111112 + y2 = 0.1111111111 + 0.1111111112 Reorder the terms: 0.1111111112 + -0.6666666667y + y2 = 0.1111111111 + 0.1111111112 Combine like terms: 0.1111111111 + 0.1111111112 = 0.2222222223 0.1111111112 + -0.6666666667y + y2 = 0.2222222223 Factor a perfect square on the left side: (y + -0.3333333334)(y + -0.3333333334) = 0.2222222223 Calculate the square root of the right side: 0.471404521 Break this problem into two subproblems by setting (y + -0.3333333334) equal to 0.471404521 and -0.471404521.Subproblem 1
y + -0.3333333334 = 0.471404521 Simplifying y + -0.3333333334 = 0.471404521 Reorder the terms: -0.3333333334 + y = 0.471404521 Solving -0.3333333334 + y = 0.471404521 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.3333333334' to each side of the equation. -0.3333333334 + 0.3333333334 + y = 0.471404521 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + y = 0.471404521 + 0.3333333334 y = 0.471404521 + 0.3333333334 Combine like terms: 0.471404521 + 0.3333333334 = 0.8047378544 y = 0.8047378544 Simplifying y = 0.8047378544Subproblem 2
y + -0.3333333334 = -0.471404521 Simplifying y + -0.3333333334 = -0.471404521 Reorder the terms: -0.3333333334 + y = -0.471404521 Solving -0.3333333334 + y = -0.471404521 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.3333333334' to each side of the equation. -0.3333333334 + 0.3333333334 + y = -0.471404521 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + y = -0.471404521 + 0.3333333334 y = -0.471404521 + 0.3333333334 Combine like terms: -0.471404521 + 0.3333333334 = -0.1380711876 y = -0.1380711876 Simplifying y = -0.1380711876Solution
The solution to the problem is based on the solutions from the subproblems. y = {0.8047378544, -0.1380711876}
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