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Simplifying 9y2 + 1 = 12y Reorder the terms: 1 + 9y2 = 12y Solving 1 + 9y2 = 12y Solving for variable 'y'. Reorder the terms: 1 + -12y + 9y2 = 12y + -12y Combine like terms: 12y + -12y = 0 1 + -12y + 9y2 = 0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. 0.1111111111 + -1.333333333y + y2 = 0 Move the constant term to the right: Add '-0.1111111111' to each side of the equation. 0.1111111111 + -1.333333333y + -0.1111111111 + y2 = 0 + -0.1111111111 Reorder the terms: 0.1111111111 + -0.1111111111 + -1.333333333y + y2 = 0 + -0.1111111111 Combine like terms: 0.1111111111 + -0.1111111111 = 0.0000000000 0.0000000000 + -1.333333333y + y2 = 0 + -0.1111111111 -1.333333333y + y2 = 0 + -0.1111111111 Combine like terms: 0 + -0.1111111111 = -0.1111111111 -1.333333333y + y2 = -0.1111111111 The y term is -1.333333333y. Take half its coefficient (-0.6666666665). Square it (0.4444444442) and add it to both sides. Add '0.4444444442' to each side of the equation. -1.333333333y + 0.4444444442 + y2 = -0.1111111111 + 0.4444444442 Reorder the terms: 0.4444444442 + -1.333333333y + y2 = -0.1111111111 + 0.4444444442 Combine like terms: -0.1111111111 + 0.4444444442 = 0.3333333331 0.4444444442 + -1.333333333y + y2 = 0.3333333331 Factor a perfect square on the left side: (y + -0.6666666665)(y + -0.6666666665) = 0.3333333331 Calculate the square root of the right side: 0.577350269 Break this problem into two subproblems by setting (y + -0.6666666665) equal to 0.577350269 and -0.577350269.Subproblem 1
y + -0.6666666665 = 0.577350269 Simplifying y + -0.6666666665 = 0.577350269 Reorder the terms: -0.6666666665 + y = 0.577350269 Solving -0.6666666665 + y = 0.577350269 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.6666666665' to each side of the equation. -0.6666666665 + 0.6666666665 + y = 0.577350269 + 0.6666666665 Combine like terms: -0.6666666665 + 0.6666666665 = 0.0000000000 0.0000000000 + y = 0.577350269 + 0.6666666665 y = 0.577350269 + 0.6666666665 Combine like terms: 0.577350269 + 0.6666666665 = 1.2440169355 y = 1.2440169355 Simplifying y = 1.2440169355Subproblem 2
y + -0.6666666665 = -0.577350269 Simplifying y + -0.6666666665 = -0.577350269 Reorder the terms: -0.6666666665 + y = -0.577350269 Solving -0.6666666665 + y = -0.577350269 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.6666666665' to each side of the equation. -0.6666666665 + 0.6666666665 + y = -0.577350269 + 0.6666666665 Combine like terms: -0.6666666665 + 0.6666666665 = 0.0000000000 0.0000000000 + y = -0.577350269 + 0.6666666665 y = -0.577350269 + 0.6666666665 Combine like terms: -0.577350269 + 0.6666666665 = 0.0893163975 y = 0.0893163975 Simplifying y = 0.0893163975Solution
The solution to the problem is based on the solutions from the subproblems. y = {1.2440169355, 0.0893163975}
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