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Simplifying (-3y2 + -7y) + (9y2 + y + -5) = 0 Reorder the terms: (-7y + -3y2) + (9y2 + y + -5) = 0 Remove parenthesis around (-7y + -3y2) -7y + -3y2 + (9y2 + y + -5) = 0 Reorder the terms: -7y + -3y2 + (-5 + y + 9y2) = 0 Remove parenthesis around (-5 + y + 9y2) -7y + -3y2 + -5 + y + 9y2 = 0 Reorder the terms: -5 + -7y + y + -3y2 + 9y2 = 0 Combine like terms: -7y + y = -6y -5 + -6y + -3y2 + 9y2 = 0 Combine like terms: -3y2 + 9y2 = 6y2 -5 + -6y + 6y2 = 0 Solving -5 + -6y + 6y2 = 0 Solving for variable 'y'. Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -0.8333333333 + -1y + y2 = 0 Move the constant term to the right: Add '0.8333333333' to each side of the equation. -0.8333333333 + -1y + 0.8333333333 + y2 = 0 + 0.8333333333 Reorder the terms: -0.8333333333 + 0.8333333333 + -1y + y2 = 0 + 0.8333333333 Combine like terms: -0.8333333333 + 0.8333333333 = 0.0000000000 0.0000000000 + -1y + y2 = 0 + 0.8333333333 -1y + y2 = 0 + 0.8333333333 Combine like terms: 0 + 0.8333333333 = 0.8333333333 -1y + y2 = 0.8333333333 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 = 0.8333333333 + 0.25 Reorder the terms: 0.25 + -1y + y2 = 0.8333333333 + 0.25 Combine like terms: 0.8333333333 + 0.25 = 1.0833333333 0.25 + -1y + y2 = 1.0833333333 Factor a perfect square on the left side: (y + -0.5)(y + -0.5) = 1.0833333333 Calculate the square root of the right side: 1.040833 Break this problem into two subproblems by setting (y + -0.5) equal to 1.040833 and -1.040833.Subproblem 1
y + -0.5 = 1.040833 Simplifying y + -0.5 = 1.040833 Reorder the terms: -0.5 + y = 1.040833 Solving -0.5 + y = 1.040833 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 = 1.040833 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + y = 1.040833 + 0.5 y = 1.040833 + 0.5 Combine like terms: 1.040833 + 0.5 = 1.540833 y = 1.540833 Simplifying y = 1.540833Subproblem 2
y + -0.5 = -1.040833 Simplifying y + -0.5 = -1.040833 Reorder the terms: -0.5 + y = -1.040833 Solving -0.5 + y = -1.040833 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 = -1.040833 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + y = -1.040833 + 0.5 y = -1.040833 + 0.5 Combine like terms: -1.040833 + 0.5 = -0.540833 y = -0.540833 Simplifying y = -0.540833Solution
The solution to the problem is based on the solutions from the subproblems. y = {1.540833, -0.540833}
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