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Simplifying 3(3y + -1) * 3(2y) = 1 Reorder the terms: 3(-1 + 3y) * 3(2y) = 1 Remove parenthesis around (2y) 3(-1 + 3y) * 3 * 2y = 1 Reorder the terms for easier multiplication: 3 * 3 * 2y(-1 + 3y) = 1 Multiply 3 * 3 9 * 2y(-1 + 3y) = 1 Multiply 9 * 2 18y(-1 + 3y) = 1 (-1 * 18y + 3y * 18y) = 1 (-18y + 54y2) = 1 Solving -18y + 54y2 = 1 Solving for variable 'y'. Reorder the terms: -1 + -18y + 54y2 = 1 + -1 Combine like terms: 1 + -1 = 0 -1 + -18y + 54y2 = 0 Begin completing the square. Divide all terms by 54 the coefficient of the squared term: Divide each side by '54'. -0.01851851852 + -0.3333333333y + y2 = 0 Move the constant term to the right: Add '0.01851851852' to each side of the equation. -0.01851851852 + -0.3333333333y + 0.01851851852 + y2 = 0 + 0.01851851852 Reorder the terms: -0.01851851852 + 0.01851851852 + -0.3333333333y + y2 = 0 + 0.01851851852 Combine like terms: -0.01851851852 + 0.01851851852 = 0.00000000000 0.00000000000 + -0.3333333333y + y2 = 0 + 0.01851851852 -0.3333333333y + y2 = 0 + 0.01851851852 Combine like terms: 0 + 0.01851851852 = 0.01851851852 -0.3333333333y + y2 = 0.01851851852 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 = 0.01851851852 + 0.02777777779 Reorder the terms: 0.02777777779 + -0.3333333333y + y2 = 0.01851851852 + 0.02777777779 Combine like terms: 0.01851851852 + 0.02777777779 = 0.04629629631 0.02777777779 + -0.3333333333y + y2 = 0.04629629631 Factor a perfect square on the left side: (y + -0.1666666667)(y + -0.1666666667) = 0.04629629631 Calculate the square root of the right side: 0.215165741 Break this problem into two subproblems by setting (y + -0.1666666667) equal to 0.215165741 and -0.215165741.Subproblem 1
y + -0.1666666667 = 0.215165741 Simplifying y + -0.1666666667 = 0.215165741 Reorder the terms: -0.1666666667 + y = 0.215165741 Solving -0.1666666667 + y = 0.215165741 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 = 0.215165741 + 0.1666666667 Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000 0.0000000000 + y = 0.215165741 + 0.1666666667 y = 0.215165741 + 0.1666666667 Combine like terms: 0.215165741 + 0.1666666667 = 0.3818324077 y = 0.3818324077 Simplifying y = 0.3818324077Subproblem 2
y + -0.1666666667 = -0.215165741 Simplifying y + -0.1666666667 = -0.215165741 Reorder the terms: -0.1666666667 + y = -0.215165741 Solving -0.1666666667 + y = -0.215165741 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 = -0.215165741 + 0.1666666667 Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000 0.0000000000 + y = -0.215165741 + 0.1666666667 y = -0.215165741 + 0.1666666667 Combine like terms: -0.215165741 + 0.1666666667 = -0.0484990743 y = -0.0484990743 Simplifying y = -0.0484990743Solution
The solution to the problem is based on the solutions from the subproblems. y = {0.3818324077, -0.0484990743}
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