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Simplifying 9y2 + -3y = 1 Reorder the terms: -3y + 9y2 = 1 Solving -3y + 9y2 = 1 Solving for variable 'y'. Reorder the terms: -1 + -3y + 9y2 = 1 + -1 Combine like terms: 1 + -1 = 0 -1 + -3y + 9y2 = 0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -0.1111111111 + -0.3333333333y + y2 = 0 Move the constant term to the right: Add '0.1111111111' to each side of the equation. -0.1111111111 + -0.3333333333y + 0.1111111111 + y2 = 0 + 0.1111111111 Reorder the terms: -0.1111111111 + 0.1111111111 + -0.3333333333y + y2 = 0 + 0.1111111111 Combine like terms: -0.1111111111 + 0.1111111111 = 0.0000000000 0.0000000000 + -0.3333333333y + y2 = 0 + 0.1111111111 -0.3333333333y + y2 = 0 + 0.1111111111 Combine like terms: 0 + 0.1111111111 = 0.1111111111 -0.3333333333y + y2 = 0.1111111111 The y term is -0.3333333333y. Take half its coefficient (-0.1666666667). Square it (0.02777777779) and add it to both sides. Add '0.02777777779' to each side of the equation. -0.3333333333y + 0.02777777779 + y2 = 0.1111111111 + 0.02777777779 Reorder the terms: 0.02777777779 + -0.3333333333y + y2 = 0.1111111111 + 0.02777777779 Combine like terms: 0.1111111111 + 0.02777777779 = 0.13888888889 0.02777777779 + -0.3333333333y + y2 = 0.13888888889 Factor a perfect square on the left side: (y + -0.1666666667)(y + -0.1666666667) = 0.13888888889 Calculate the square root of the right side: 0.372677996 Break this problem into two subproblems by setting (y + -0.1666666667) equal to 0.372677996 and -0.372677996.Subproblem 1
y + -0.1666666667 = 0.372677996 Simplifying y + -0.1666666667 = 0.372677996 Reorder the terms: -0.1666666667 + y = 0.372677996 Solving -0.1666666667 + y = 0.372677996 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.1666666667' to each side of the equation. -0.1666666667 + 0.1666666667 + y = 0.372677996 + 0.1666666667 Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000 0.0000000000 + y = 0.372677996 + 0.1666666667 y = 0.372677996 + 0.1666666667 Combine like terms: 0.372677996 + 0.1666666667 = 0.5393446627 y = 0.5393446627 Simplifying y = 0.5393446627Subproblem 2
y + -0.1666666667 = -0.372677996 Simplifying y + -0.1666666667 = -0.372677996 Reorder the terms: -0.1666666667 + y = -0.372677996 Solving -0.1666666667 + y = -0.372677996 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.1666666667' to each side of the equation. -0.1666666667 + 0.1666666667 + y = -0.372677996 + 0.1666666667 Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000 0.0000000000 + y = -0.372677996 + 0.1666666667 y = -0.372677996 + 0.1666666667 Combine like terms: -0.372677996 + 0.1666666667 = -0.2060113293 y = -0.2060113293 Simplifying y = -0.2060113293Solution
The solution to the problem is based on the solutions from the subproblems. y = {0.5393446627, -0.2060113293}
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