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Simplifying 5(y + 2) + -4y = 8y(2 + y) Reorder the terms: 5(2 + y) + -4y = 8y(2 + y) (2 * 5 + y * 5) + -4y = 8y(2 + y) (10 + 5y) + -4y = 8y(2 + y) Combine like terms: 5y + -4y = 1y 10 + 1y = 8y(2 + y) 10 + 1y = (2 * 8y + y * 8y) 10 + 1y = (16y + 8y2) Solving 10 + 1y = 16y + 8y2 Solving for variable 'y'. Combine like terms: 1y + -16y = -15y 10 + -15y + -8y2 = 16y + 8y2 + -16y + -8y2 Reorder the terms: 10 + -15y + -8y2 = 16y + -16y + 8y2 + -8y2 Combine like terms: 16y + -16y = 0 10 + -15y + -8y2 = 0 + 8y2 + -8y2 10 + -15y + -8y2 = 8y2 + -8y2 Combine like terms: 8y2 + -8y2 = 0 10 + -15y + -8y2 = 0 Begin completing the square. Divide all terms by -8 the coefficient of the squared term: Divide each side by '-8'. -1.25 + 1.875y + y2 = 0 Move the constant term to the right: Add '1.25' to each side of the equation. -1.25 + 1.875y + 1.25 + y2 = 0 + 1.25 Reorder the terms: -1.25 + 1.25 + 1.875y + y2 = 0 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + 1.875y + y2 = 0 + 1.25 1.875y + y2 = 0 + 1.25 Combine like terms: 0 + 1.25 = 1.25 1.875y + y2 = 1.25 The y term is 1.875y. Take half its coefficient (0.9375). Square it (0.87890625) and add it to both sides. Add '0.87890625' to each side of the equation. 1.875y + 0.87890625 + y2 = 1.25 + 0.87890625 Reorder the terms: 0.87890625 + 1.875y + y2 = 1.25 + 0.87890625 Combine like terms: 1.25 + 0.87890625 = 2.12890625 0.87890625 + 1.875y + y2 = 2.12890625 Factor a perfect square on the left side: (y + 0.9375)(y + 0.9375) = 2.12890625 Calculate the square root of the right side: 1.459077191 Break this problem into two subproblems by setting (y + 0.9375) equal to 1.459077191 and -1.459077191.Subproblem 1
y + 0.9375 = 1.459077191 Simplifying y + 0.9375 = 1.459077191 Reorder the terms: 0.9375 + y = 1.459077191 Solving 0.9375 + y = 1.459077191 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.9375' to each side of the equation. 0.9375 + -0.9375 + y = 1.459077191 + -0.9375 Combine like terms: 0.9375 + -0.9375 = 0.0000 0.0000 + y = 1.459077191 + -0.9375 y = 1.459077191 + -0.9375 Combine like terms: 1.459077191 + -0.9375 = 0.521577191 y = 0.521577191 Simplifying y = 0.521577191Subproblem 2
y + 0.9375 = -1.459077191 Simplifying y + 0.9375 = -1.459077191 Reorder the terms: 0.9375 + y = -1.459077191 Solving 0.9375 + y = -1.459077191 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.9375' to each side of the equation. 0.9375 + -0.9375 + y = -1.459077191 + -0.9375 Combine like terms: 0.9375 + -0.9375 = 0.0000 0.0000 + y = -1.459077191 + -0.9375 y = -1.459077191 + -0.9375 Combine like terms: -1.459077191 + -0.9375 = -2.396577191 y = -2.396577191 Simplifying y = -2.396577191Solution
The solution to the problem is based on the solutions from the subproblems. y = {0.521577191, -2.396577191}
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