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Simplifying (5y + 8)(7 + 4y) = 4 Reorder the terms: (8 + 5y)(7 + 4y) = 4 Multiply (8 + 5y) * (7 + 4y) (8(7 + 4y) + 5y * (7 + 4y)) = 4 ((7 * 8 + 4y * 8) + 5y * (7 + 4y)) = 4 ((56 + 32y) + 5y * (7 + 4y)) = 4 (56 + 32y + (7 * 5y + 4y * 5y)) = 4 (56 + 32y + (35y + 20y2)) = 4 Combine like terms: 32y + 35y = 67y (56 + 67y + 20y2) = 4 Solving 56 + 67y + 20y2 = 4 Solving for variable 'y'. Reorder the terms: 56 + -4 + 67y + 20y2 = 4 + -4 Combine like terms: 56 + -4 = 52 52 + 67y + 20y2 = 4 + -4 Combine like terms: 4 + -4 = 0 52 + 67y + 20y2 = 0 Begin completing the square. Divide all terms by 20 the coefficient of the squared term: Divide each side by '20'. 2.6 + 3.35y + y2 = 0.0 Move the constant term to the right: Add '-2.6' to each side of the equation. 2.6 + 3.35y + -2.6 + y2 = 0.0 + -2.6 Reorder the terms: 2.6 + -2.6 + 3.35y + y2 = 0.0 + -2.6 Combine like terms: 2.6 + -2.6 = 0.0 0.0 + 3.35y + y2 = 0.0 + -2.6 3.35y + y2 = 0.0 + -2.6 Combine like terms: 0.0 + -2.6 = -2.6 3.35y + y2 = -2.6 The y term is 3.35y. Take half its coefficient (1.675). Square it (2.805625) and add it to both sides. Add '2.805625' to each side of the equation. 3.35y + 2.805625 + y2 = -2.6 + 2.805625 Reorder the terms: 2.805625 + 3.35y + y2 = -2.6 + 2.805625 Combine like terms: -2.6 + 2.805625 = 0.205625 2.805625 + 3.35y + y2 = 0.205625 Factor a perfect square on the left side: (y + 1.675)(y + 1.675) = 0.205625 Calculate the square root of the right side: 0.453458929 Break this problem into two subproblems by setting (y + 1.675) equal to 0.453458929 and -0.453458929.Subproblem 1
y + 1.675 = 0.453458929 Simplifying y + 1.675 = 0.453458929 Reorder the terms: 1.675 + y = 0.453458929 Solving 1.675 + y = 0.453458929 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.675' to each side of the equation. 1.675 + -1.675 + y = 0.453458929 + -1.675 Combine like terms: 1.675 + -1.675 = 0.000 0.000 + y = 0.453458929 + -1.675 y = 0.453458929 + -1.675 Combine like terms: 0.453458929 + -1.675 = -1.221541071 y = -1.221541071 Simplifying y = -1.221541071Subproblem 2
y + 1.675 = -0.453458929 Simplifying y + 1.675 = -0.453458929 Reorder the terms: 1.675 + y = -0.453458929 Solving 1.675 + y = -0.453458929 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.675' to each side of the equation. 1.675 + -1.675 + y = -0.453458929 + -1.675 Combine like terms: 1.675 + -1.675 = 0.000 0.000 + y = -0.453458929 + -1.675 y = -0.453458929 + -1.675 Combine like terms: -0.453458929 + -1.675 = -2.128458929 y = -2.128458929 Simplifying y = -2.128458929Solution
The solution to the problem is based on the solutions from the subproblems. y = {-1.221541071, -2.128458929}
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