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Simplifying (3y + 4)(-2y + 6) + (3y + 6) = 0 Reorder the terms: (4 + 3y)(-2y + 6) + (3y + 6) = 0 Reorder the terms: (4 + 3y)(6 + -2y) + (3y + 6) = 0 Multiply (4 + 3y) * (6 + -2y) (4(6 + -2y) + 3y * (6 + -2y)) + (3y + 6) = 0 ((6 * 4 + -2y * 4) + 3y * (6 + -2y)) + (3y + 6) = 0 ((24 + -8y) + 3y * (6 + -2y)) + (3y + 6) = 0 (24 + -8y + (6 * 3y + -2y * 3y)) + (3y + 6) = 0 (24 + -8y + (18y + -6y2)) + (3y + 6) = 0 Combine like terms: -8y + 18y = 10y (24 + 10y + -6y2) + (3y + 6) = 0 Reorder the terms: 24 + 10y + -6y2 + (6 + 3y) = 0 Remove parenthesis around (6 + 3y) 24 + 10y + -6y2 + 6 + 3y = 0 Reorder the terms: 24 + 6 + 10y + 3y + -6y2 = 0 Combine like terms: 24 + 6 = 30 30 + 10y + 3y + -6y2 = 0 Combine like terms: 10y + 3y = 13y 30 + 13y + -6y2 = 0 Solving 30 + 13y + -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'. -5 + -2.166666667y + y2 = 0 Move the constant term to the right: Add '5' to each side of the equation. -5 + -2.166666667y + 5 + y2 = 0 + 5 Reorder the terms: -5 + 5 + -2.166666667y + y2 = 0 + 5 Combine like terms: -5 + 5 = 0 0 + -2.166666667y + y2 = 0 + 5 -2.166666667y + y2 = 0 + 5 Combine like terms: 0 + 5 = 5 -2.166666667y + y2 = 5 The y term is -2.166666667y. Take half its coefficient (-1.083333334). Square it (1.173611113) and add it to both sides. Add '1.173611113' to each side of the equation. -2.166666667y + 1.173611113 + y2 = 5 + 1.173611113 Reorder the terms: 1.173611113 + -2.166666667y + y2 = 5 + 1.173611113 Combine like terms: 5 + 1.173611113 = 6.173611113 1.173611113 + -2.166666667y + y2 = 6.173611113 Factor a perfect square on the left side: (y + -1.083333334)(y + -1.083333334) = 6.173611113 Calculate the square root of the right side: 2.484675253 Break this problem into two subproblems by setting (y + -1.083333334) equal to 2.484675253 and -2.484675253.Subproblem 1
y + -1.083333334 = 2.484675253 Simplifying y + -1.083333334 = 2.484675253 Reorder the terms: -1.083333334 + y = 2.484675253 Solving -1.083333334 + y = 2.484675253 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.083333334' to each side of the equation. -1.083333334 + 1.083333334 + y = 2.484675253 + 1.083333334 Combine like terms: -1.083333334 + 1.083333334 = 0.000000000 0.000000000 + y = 2.484675253 + 1.083333334 y = 2.484675253 + 1.083333334 Combine like terms: 2.484675253 + 1.083333334 = 3.568008587 y = 3.568008587 Simplifying y = 3.568008587Subproblem 2
y + -1.083333334 = -2.484675253 Simplifying y + -1.083333334 = -2.484675253 Reorder the terms: -1.083333334 + y = -2.484675253 Solving -1.083333334 + y = -2.484675253 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.083333334' to each side of the equation. -1.083333334 + 1.083333334 + y = -2.484675253 + 1.083333334 Combine like terms: -1.083333334 + 1.083333334 = 0.000000000 0.000000000 + y = -2.484675253 + 1.083333334 y = -2.484675253 + 1.083333334 Combine like terms: -2.484675253 + 1.083333334 = -1.401341919 y = -1.401341919 Simplifying y = -1.401341919Solution
The solution to the problem is based on the solutions from the subproblems. y = {3.568008587, -1.401341919}
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