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Simplifying 11y + 3 = 4y(2y + 3) Reorder the terms: 3 + 11y = 4y(2y + 3) Reorder the terms: 3 + 11y = 4y(3 + 2y) 3 + 11y = (3 * 4y + 2y * 4y) 3 + 11y = (12y + 8y2) Solving 3 + 11y = 12y + 8y2 Solving for variable 'y'. Combine like terms: 11y + -12y = -1y 3 + -1y + -8y2 = 12y + 8y2 + -12y + -8y2 Reorder the terms: 3 + -1y + -8y2 = 12y + -12y + 8y2 + -8y2 Combine like terms: 12y + -12y = 0 3 + -1y + -8y2 = 0 + 8y2 + -8y2 3 + -1y + -8y2 = 8y2 + -8y2 Combine like terms: 8y2 + -8y2 = 0 3 + -1y + -8y2 = 0 Begin completing the square. Divide all terms by -8 the coefficient of the squared term: Divide each side by '-8'. -0.375 + 0.125y + y2 = 0 Move the constant term to the right: Add '0.375' to each side of the equation. -0.375 + 0.125y + 0.375 + y2 = 0 + 0.375 Reorder the terms: -0.375 + 0.375 + 0.125y + y2 = 0 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + 0.125y + y2 = 0 + 0.375 0.125y + y2 = 0 + 0.375 Combine like terms: 0 + 0.375 = 0.375 0.125y + y2 = 0.375 The y term is 0.125y. Take half its coefficient (0.0625). Square it (0.00390625) and add it to both sides. Add '0.00390625' to each side of the equation. 0.125y + 0.00390625 + y2 = 0.375 + 0.00390625 Reorder the terms: 0.00390625 + 0.125y + y2 = 0.375 + 0.00390625 Combine like terms: 0.375 + 0.00390625 = 0.37890625 0.00390625 + 0.125y + y2 = 0.37890625 Factor a perfect square on the left side: (y + 0.0625)(y + 0.0625) = 0.37890625 Calculate the square root of the right side: 0.615553613 Break this problem into two subproblems by setting (y + 0.0625) equal to 0.615553613 and -0.615553613.Subproblem 1
y + 0.0625 = 0.615553613 Simplifying y + 0.0625 = 0.615553613 Reorder the terms: 0.0625 + y = 0.615553613 Solving 0.0625 + y = 0.615553613 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.0625' to each side of the equation. 0.0625 + -0.0625 + y = 0.615553613 + -0.0625 Combine like terms: 0.0625 + -0.0625 = 0.0000 0.0000 + y = 0.615553613 + -0.0625 y = 0.615553613 + -0.0625 Combine like terms: 0.615553613 + -0.0625 = 0.553053613 y = 0.553053613 Simplifying y = 0.553053613Subproblem 2
y + 0.0625 = -0.615553613 Simplifying y + 0.0625 = -0.615553613 Reorder the terms: 0.0625 + y = -0.615553613 Solving 0.0625 + y = -0.615553613 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.0625' to each side of the equation. 0.0625 + -0.0625 + y = -0.615553613 + -0.0625 Combine like terms: 0.0625 + -0.0625 = 0.0000 0.0000 + y = -0.615553613 + -0.0625 y = -0.615553613 + -0.0625 Combine like terms: -0.615553613 + -0.0625 = -0.678053613 y = -0.678053613 Simplifying y = -0.678053613Solution
The solution to the problem is based on the solutions from the subproblems. y = {0.553053613, -0.678053613}
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