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Simplifying 3y(3y + -1) + -3(2y) = 1 Reorder the terms: 3y(-1 + 3y) + -3(2y) = 1 (-1 * 3y + 3y * 3y) + -3(2y) = 1 (-3y + 9y2) + -3(2y) = 1 Remove parenthesis around (2y) -3y + 9y2 + -3 * 2y = 1 Multiply -3 * 2 -3y + 9y2 + -6y = 1 Reorder the terms: -3y + -6y + 9y2 = 1 Combine like terms: -3y + -6y = -9y -9y + 9y2 = 1 Solving -9y + 9y2 = 1 Solving for variable 'y'. Reorder the terms: -1 + -9y + 9y2 = 1 + -1 Combine like terms: 1 + -1 = 0 -1 + -9y + 9y2 = 0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -0.1111111111 + -1y + y2 = 0 Move the constant term to the right: Add '0.1111111111' to each side of the equation. -0.1111111111 + -1y + 0.1111111111 + y2 = 0 + 0.1111111111 Reorder the terms: -0.1111111111 + 0.1111111111 + -1y + y2 = 0 + 0.1111111111 Combine like terms: -0.1111111111 + 0.1111111111 = 0.0000000000 0.0000000000 + -1y + y2 = 0 + 0.1111111111 -1y + y2 = 0 + 0.1111111111 Combine like terms: 0 + 0.1111111111 = 0.1111111111 -1y + y2 = 0.1111111111 The y term is -1y. Take half its coefficient (-0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -1y + 0.25 + y2 = 0.1111111111 + 0.25 Reorder the terms: 0.25 + -1y + y2 = 0.1111111111 + 0.25 Combine like terms: 0.1111111111 + 0.25 = 0.3611111111 0.25 + -1y + y2 = 0.3611111111 Factor a perfect square on the left side: (y + -0.5)(y + -0.5) = 0.3611111111 Calculate the square root of the right side: 0.600925213 Break this problem into two subproblems by setting (y + -0.5) equal to 0.600925213 and -0.600925213.Subproblem 1
y + -0.5 = 0.600925213 Simplifying y + -0.5 = 0.600925213 Reorder the terms: -0.5 + y = 0.600925213 Solving -0.5 + y = 0.600925213 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + y = 0.600925213 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + y = 0.600925213 + 0.5 y = 0.600925213 + 0.5 Combine like terms: 0.600925213 + 0.5 = 1.100925213 y = 1.100925213 Simplifying y = 1.100925213Subproblem 2
y + -0.5 = -0.600925213 Simplifying y + -0.5 = -0.600925213 Reorder the terms: -0.5 + y = -0.600925213 Solving -0.5 + y = -0.600925213 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + y = -0.600925213 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + y = -0.600925213 + 0.5 y = -0.600925213 + 0.5 Combine like terms: -0.600925213 + 0.5 = -0.100925213 y = -0.100925213 Simplifying y = -0.100925213Solution
The solution to the problem is based on the solutions from the subproblems. y = {1.100925213, -0.100925213}
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