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Simplifying (3y + -1)(4y + 7) = 90 Reorder the terms: (-1 + 3y)(4y + 7) = 90 Reorder the terms: (-1 + 3y)(7 + 4y) = 90 Multiply (-1 + 3y) * (7 + 4y) (-1(7 + 4y) + 3y * (7 + 4y)) = 90 ((7 * -1 + 4y * -1) + 3y * (7 + 4y)) = 90 ((-7 + -4y) + 3y * (7 + 4y)) = 90 (-7 + -4y + (7 * 3y + 4y * 3y)) = 90 (-7 + -4y + (21y + 12y2)) = 90 Combine like terms: -4y + 21y = 17y (-7 + 17y + 12y2) = 90 Solving -7 + 17y + 12y2 = 90 Solving for variable 'y'. Reorder the terms: -7 + -90 + 17y + 12y2 = 90 + -90 Combine like terms: -7 + -90 = -97 -97 + 17y + 12y2 = 90 + -90 Combine like terms: 90 + -90 = 0 -97 + 17y + 12y2 = 0 Begin completing the square. Divide all terms by 12 the coefficient of the squared term: Divide each side by '12'. -8.083333333 + 1.416666667y + y2 = 0 Move the constant term to the right: Add '8.083333333' to each side of the equation. -8.083333333 + 1.416666667y + 8.083333333 + y2 = 0 + 8.083333333 Reorder the terms: -8.083333333 + 8.083333333 + 1.416666667y + y2 = 0 + 8.083333333 Combine like terms: -8.083333333 + 8.083333333 = 0.000000000 0.000000000 + 1.416666667y + y2 = 0 + 8.083333333 1.416666667y + y2 = 0 + 8.083333333 Combine like terms: 0 + 8.083333333 = 8.083333333 1.416666667y + y2 = 8.083333333 The y term is 1.416666667y. Take half its coefficient (0.7083333335). Square it (0.5017361113) and add it to both sides. Add '0.5017361113' to each side of the equation. 1.416666667y + 0.5017361113 + y2 = 8.083333333 + 0.5017361113 Reorder the terms: 0.5017361113 + 1.416666667y + y2 = 8.083333333 + 0.5017361113 Combine like terms: 8.083333333 + 0.5017361113 = 8.5850694443 0.5017361113 + 1.416666667y + y2 = 8.5850694443 Factor a perfect square on the left side: (y + 0.7083333335)(y + 0.7083333335) = 8.5850694443 Calculate the square root of the right side: 2.930028915 Break this problem into two subproblems by setting (y + 0.7083333335) equal to 2.930028915 and -2.930028915.Subproblem 1
y + 0.7083333335 = 2.930028915 Simplifying y + 0.7083333335 = 2.930028915 Reorder the terms: 0.7083333335 + y = 2.930028915 Solving 0.7083333335 + y = 2.930028915 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.7083333335' to each side of the equation. 0.7083333335 + -0.7083333335 + y = 2.930028915 + -0.7083333335 Combine like terms: 0.7083333335 + -0.7083333335 = 0.0000000000 0.0000000000 + y = 2.930028915 + -0.7083333335 y = 2.930028915 + -0.7083333335 Combine like terms: 2.930028915 + -0.7083333335 = 2.2216955815 y = 2.2216955815 Simplifying y = 2.2216955815Subproblem 2
y + 0.7083333335 = -2.930028915 Simplifying y + 0.7083333335 = -2.930028915 Reorder the terms: 0.7083333335 + y = -2.930028915 Solving 0.7083333335 + y = -2.930028915 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.7083333335' to each side of the equation. 0.7083333335 + -0.7083333335 + y = -2.930028915 + -0.7083333335 Combine like terms: 0.7083333335 + -0.7083333335 = 0.0000000000 0.0000000000 + y = -2.930028915 + -0.7083333335 y = -2.930028915 + -0.7083333335 Combine like terms: -2.930028915 + -0.7083333335 = -3.6383622485 y = -3.6383622485 Simplifying y = -3.6383622485Solution
The solution to the problem is based on the solutions from the subproblems. y = {2.2216955815, -3.6383622485}
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