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Simplifying 4(y + 4) * y = 3(y + 2) + -4 Reorder the terms: 4(4 + y) * y = 3(y + 2) + -4 Reorder the terms for easier multiplication: 4y(4 + y) = 3(y + 2) + -4 (4 * 4y + y * 4y) = 3(y + 2) + -4 (16y + 4y2) = 3(y + 2) + -4 Reorder the terms: 16y + 4y2 = 3(2 + y) + -4 16y + 4y2 = (2 * 3 + y * 3) + -4 16y + 4y2 = (6 + 3y) + -4 Reorder the terms: 16y + 4y2 = 6 + -4 + 3y Combine like terms: 6 + -4 = 2 16y + 4y2 = 2 + 3y Solving 16y + 4y2 = 2 + 3y Solving for variable 'y'. Reorder the terms: -2 + 16y + -3y + 4y2 = 2 + 3y + -2 + -3y Combine like terms: 16y + -3y = 13y -2 + 13y + 4y2 = 2 + 3y + -2 + -3y Reorder the terms: -2 + 13y + 4y2 = 2 + -2 + 3y + -3y Combine like terms: 2 + -2 = 0 -2 + 13y + 4y2 = 0 + 3y + -3y -2 + 13y + 4y2 = 3y + -3y Combine like terms: 3y + -3y = 0 -2 + 13y + 4y2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -0.5 + 3.25y + y2 = 0 Move the constant term to the right: Add '0.5' to each side of the equation. -0.5 + 3.25y + 0.5 + y2 = 0 + 0.5 Reorder the terms: -0.5 + 0.5 + 3.25y + y2 = 0 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + 3.25y + y2 = 0 + 0.5 3.25y + y2 = 0 + 0.5 Combine like terms: 0 + 0.5 = 0.5 3.25y + y2 = 0.5 The y term is 3.25y. Take half its coefficient (1.625). Square it (2.640625) and add it to both sides. Add '2.640625' to each side of the equation. 3.25y + 2.640625 + y2 = 0.5 + 2.640625 Reorder the terms: 2.640625 + 3.25y + y2 = 0.5 + 2.640625 Combine like terms: 0.5 + 2.640625 = 3.140625 2.640625 + 3.25y + y2 = 3.140625 Factor a perfect square on the left side: (y + 1.625)(y + 1.625) = 3.140625 Calculate the square root of the right side: 1.77218086 Break this problem into two subproblems by setting (y + 1.625) equal to 1.77218086 and -1.77218086.Subproblem 1
y + 1.625 = 1.77218086 Simplifying y + 1.625 = 1.77218086 Reorder the terms: 1.625 + y = 1.77218086 Solving 1.625 + y = 1.77218086 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.625' to each side of the equation. 1.625 + -1.625 + y = 1.77218086 + -1.625 Combine like terms: 1.625 + -1.625 = 0.000 0.000 + y = 1.77218086 + -1.625 y = 1.77218086 + -1.625 Combine like terms: 1.77218086 + -1.625 = 0.14718086 y = 0.14718086 Simplifying y = 0.14718086Subproblem 2
y + 1.625 = -1.77218086 Simplifying y + 1.625 = -1.77218086 Reorder the terms: 1.625 + y = -1.77218086 Solving 1.625 + y = -1.77218086 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.625' to each side of the equation. 1.625 + -1.625 + y = -1.77218086 + -1.625 Combine like terms: 1.625 + -1.625 = 0.000 0.000 + y = -1.77218086 + -1.625 y = -1.77218086 + -1.625 Combine like terms: -1.77218086 + -1.625 = -3.39718086 y = -3.39718086 Simplifying y = -3.39718086Solution
The solution to the problem is based on the solutions from the subproblems. y = {0.14718086, -3.39718086}
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