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Simplifying (4y + 7)(9y + 4) = 180 Reorder the terms: (7 + 4y)(9y + 4) = 180 Reorder the terms: (7 + 4y)(4 + 9y) = 180 Multiply (7 + 4y) * (4 + 9y) (7(4 + 9y) + 4y * (4 + 9y)) = 180 ((4 * 7 + 9y * 7) + 4y * (4 + 9y)) = 180 ((28 + 63y) + 4y * (4 + 9y)) = 180 (28 + 63y + (4 * 4y + 9y * 4y)) = 180 (28 + 63y + (16y + 36y2)) = 180 Combine like terms: 63y + 16y = 79y (28 + 79y + 36y2) = 180 Solving 28 + 79y + 36y2 = 180 Solving for variable 'y'. Reorder the terms: 28 + -180 + 79y + 36y2 = 180 + -180 Combine like terms: 28 + -180 = -152 -152 + 79y + 36y2 = 180 + -180 Combine like terms: 180 + -180 = 0 -152 + 79y + 36y2 = 0 Begin completing the square. Divide all terms by 36 the coefficient of the squared term: Divide each side by '36'. -4.222222222 + 2.194444444y + y2 = 0 Move the constant term to the right: Add '4.222222222' to each side of the equation. -4.222222222 + 2.194444444y + 4.222222222 + y2 = 0 + 4.222222222 Reorder the terms: -4.222222222 + 4.222222222 + 2.194444444y + y2 = 0 + 4.222222222 Combine like terms: -4.222222222 + 4.222222222 = 0.000000000 0.000000000 + 2.194444444y + y2 = 0 + 4.222222222 2.194444444y + y2 = 0 + 4.222222222 Combine like terms: 0 + 4.222222222 = 4.222222222 2.194444444y + y2 = 4.222222222 The y term is 2.194444444y. Take half its coefficient (1.097222222). Square it (1.203896604) and add it to both sides. Add '1.203896604' to each side of the equation. 2.194444444y + 1.203896604 + y2 = 4.222222222 + 1.203896604 Reorder the terms: 1.203896604 + 2.194444444y + y2 = 4.222222222 + 1.203896604 Combine like terms: 4.222222222 + 1.203896604 = 5.426118826 1.203896604 + 2.194444444y + y2 = 5.426118826 Factor a perfect square on the left side: (y + 1.097222222)(y + 1.097222222) = 5.426118826 Calculate the square root of the right side: 2.329403105 Break this problem into two subproblems by setting (y + 1.097222222) equal to 2.329403105 and -2.329403105.Subproblem 1
y + 1.097222222 = 2.329403105 Simplifying y + 1.097222222 = 2.329403105 Reorder the terms: 1.097222222 + y = 2.329403105 Solving 1.097222222 + y = 2.329403105 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.097222222' to each side of the equation. 1.097222222 + -1.097222222 + y = 2.329403105 + -1.097222222 Combine like terms: 1.097222222 + -1.097222222 = 0.000000000 0.000000000 + y = 2.329403105 + -1.097222222 y = 2.329403105 + -1.097222222 Combine like terms: 2.329403105 + -1.097222222 = 1.232180883 y = 1.232180883 Simplifying y = 1.232180883Subproblem 2
y + 1.097222222 = -2.329403105 Simplifying y + 1.097222222 = -2.329403105 Reorder the terms: 1.097222222 + y = -2.329403105 Solving 1.097222222 + y = -2.329403105 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.097222222' to each side of the equation. 1.097222222 + -1.097222222 + y = -2.329403105 + -1.097222222 Combine like terms: 1.097222222 + -1.097222222 = 0.000000000 0.000000000 + y = -2.329403105 + -1.097222222 y = -2.329403105 + -1.097222222 Combine like terms: -2.329403105 + -1.097222222 = -3.426625327 y = -3.426625327 Simplifying y = -3.426625327Solution
The solution to the problem is based on the solutions from the subproblems. y = {1.232180883, -3.426625327}
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