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Simplifying -6y2 + -9y = 1 Reorder the terms: -9y + -6y2 = 1 Solving -9y + -6y2 = 1 Solving for variable 'y'. Reorder the terms: -1 + -9y + -6y2 = 1 + -1 Combine like terms: 1 + -1 = 0 -1 + -9y + -6y2 = 0 Factor out the Greatest Common Factor (GCF), '-1'. -1(1 + 9y + 6y2) = 0 Ignore the factor -1.Subproblem 1
Set the factor '(1 + 9y + 6y2)' equal to zero and attempt to solve: Simplifying 1 + 9y + 6y2 = 0 Solving 1 + 9y + 6y2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. 0.1666666667 + 1.5y + y2 = 0 Move the constant term to the right: Add '-0.1666666667' to each side of the equation. 0.1666666667 + 1.5y + -0.1666666667 + y2 = 0 + -0.1666666667 Reorder the terms: 0.1666666667 + -0.1666666667 + 1.5y + y2 = 0 + -0.1666666667 Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000 0.0000000000 + 1.5y + y2 = 0 + -0.1666666667 1.5y + y2 = 0 + -0.1666666667 Combine like terms: 0 + -0.1666666667 = -0.1666666667 1.5y + y2 = -0.1666666667 The y term is 1.5y. Take half its coefficient (0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. 1.5y + 0.5625 + y2 = -0.1666666667 + 0.5625 Reorder the terms: 0.5625 + 1.5y + y2 = -0.1666666667 + 0.5625 Combine like terms: -0.1666666667 + 0.5625 = 0.3958333333 0.5625 + 1.5y + y2 = 0.3958333333 Factor a perfect square on the left side: (y + 0.75)(y + 0.75) = 0.3958333333 Calculate the square root of the right side: 0.62915287 Break this problem into two subproblems by setting (y + 0.75) equal to 0.62915287 and -0.62915287.Subproblem 1
y + 0.75 = 0.62915287 Simplifying y + 0.75 = 0.62915287 Reorder the terms: 0.75 + y = 0.62915287 Solving 0.75 + y = 0.62915287 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + y = 0.62915287 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + y = 0.62915287 + -0.75 y = 0.62915287 + -0.75 Combine like terms: 0.62915287 + -0.75 = -0.12084713 y = -0.12084713 Simplifying y = -0.12084713Subproblem 2
y + 0.75 = -0.62915287 Simplifying y + 0.75 = -0.62915287 Reorder the terms: 0.75 + y = -0.62915287 Solving 0.75 + y = -0.62915287 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + y = -0.62915287 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + y = -0.62915287 + -0.75 y = -0.62915287 + -0.75 Combine like terms: -0.62915287 + -0.75 = -1.37915287 y = -1.37915287 Simplifying y = -1.37915287Solution
The solution to the problem is based on the solutions from the subproblems. y = {-0.12084713, -1.37915287}Solution
y = {-0.12084713, -1.37915287}
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