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Simplifying (5y2 + -7y) + (4y2 + y + -2) = 0 Reorder the terms: (-7y + 5y2) + (4y2 + y + -2) = 0 Remove parenthesis around (-7y + 5y2) -7y + 5y2 + (4y2 + y + -2) = 0 Reorder the terms: -7y + 5y2 + (-2 + y + 4y2) = 0 Remove parenthesis around (-2 + y + 4y2) -7y + 5y2 + -2 + y + 4y2 = 0 Reorder the terms: -2 + -7y + y + 5y2 + 4y2 = 0 Combine like terms: -7y + y = -6y -2 + -6y + 5y2 + 4y2 = 0 Combine like terms: 5y2 + 4y2 = 9y2 -2 + -6y + 9y2 = 0 Solving -2 + -6y + 9y2 = 0 Solving for variable 'y'. Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -0.2222222222 + -0.6666666667y + y2 = 0 Move the constant term to the right: Add '0.2222222222' to each side of the equation. -0.2222222222 + -0.6666666667y + 0.2222222222 + y2 = 0 + 0.2222222222 Reorder the terms: -0.2222222222 + 0.2222222222 + -0.6666666667y + y2 = 0 + 0.2222222222 Combine like terms: -0.2222222222 + 0.2222222222 = 0.0000000000 0.0000000000 + -0.6666666667y + y2 = 0 + 0.2222222222 -0.6666666667y + y2 = 0 + 0.2222222222 Combine like terms: 0 + 0.2222222222 = 0.2222222222 -0.6666666667y + y2 = 0.2222222222 The y term is -0.6666666667y. Take half its coefficient (-0.3333333334). Square it (0.1111111112) and add it to both sides. Add '0.1111111112' to each side of the equation. -0.6666666667y + 0.1111111112 + y2 = 0.2222222222 + 0.1111111112 Reorder the terms: 0.1111111112 + -0.6666666667y + y2 = 0.2222222222 + 0.1111111112 Combine like terms: 0.2222222222 + 0.1111111112 = 0.3333333334 0.1111111112 + -0.6666666667y + y2 = 0.3333333334 Factor a perfect square on the left side: (y + -0.3333333334)(y + -0.3333333334) = 0.3333333334 Calculate the square root of the right side: 0.577350269 Break this problem into two subproblems by setting (y + -0.3333333334) equal to 0.577350269 and -0.577350269.Subproblem 1
y + -0.3333333334 = 0.577350269 Simplifying y + -0.3333333334 = 0.577350269 Reorder the terms: -0.3333333334 + y = 0.577350269 Solving -0.3333333334 + y = 0.577350269 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.3333333334' to each side of the equation. -0.3333333334 + 0.3333333334 + y = 0.577350269 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + y = 0.577350269 + 0.3333333334 y = 0.577350269 + 0.3333333334 Combine like terms: 0.577350269 + 0.3333333334 = 0.9106836024 y = 0.9106836024 Simplifying y = 0.9106836024Subproblem 2
y + -0.3333333334 = -0.577350269 Simplifying y + -0.3333333334 = -0.577350269 Reorder the terms: -0.3333333334 + y = -0.577350269 Solving -0.3333333334 + y = -0.577350269 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.3333333334' to each side of the equation. -0.3333333334 + 0.3333333334 + y = -0.577350269 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + y = -0.577350269 + 0.3333333334 y = -0.577350269 + 0.3333333334 Combine like terms: -0.577350269 + 0.3333333334 = -0.2440169356 y = -0.2440169356 Simplifying y = -0.2440169356Solution
The solution to the problem is based on the solutions from the subproblems. y = {0.9106836024, -0.2440169356}
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