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Simplifying 8y(2y + -3) = 9 Reorder the terms: 8y(-3 + 2y) = 9 (-3 * 8y + 2y * 8y) = 9 (-24y + 16y^{2}) = 9 Solving -24y + 16y^{2}= 9 Solving for variable 'y'. Reorder the terms: -9 + -24y + 16y^{2}= 9 + -9 Combine like terms: 9 + -9 = 0 -9 + -24y + 16y^{2}= 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -0.5625 + -1.5y + y^{2}= 0 Move the constant term to the right: Add '0.5625' to each side of the equation. -0.5625 + -1.5y + 0.5625 + y^{2}= 0 + 0.5625 Reorder the terms: -0.5625 + 0.5625 + -1.5y + y^{2}= 0 + 0.5625 Combine like terms: -0.5625 + 0.5625 = 0.0000 0.0000 + -1.5y + y^{2}= 0 + 0.5625 -1.5y + y^{2}= 0 + 0.5625 Combine like terms: 0 + 0.5625 = 0.5625 -1.5y + y^{2}= 0.5625 The y term is -1.5y. Take half its coefficient (-0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. -1.5y + 0.5625 + y^{2}= 0.5625 + 0.5625 Reorder the terms: 0.5625 + -1.5y + y^{2}= 0.5625 + 0.5625 Combine like terms: 0.5625 + 0.5625 = 1.125 0.5625 + -1.5y + y^{2}= 1.125 Factor a perfect square on the left side: (y + -0.75)(y + -0.75) = 1.125 Calculate the square root of the right side: 1.060660172 Break this problem into two subproblems by setting (y + -0.75) equal to 1.060660172 and -1.060660172.## Subproblem 1

y + -0.75 = 1.060660172 Simplifying y + -0.75 = 1.060660172 Reorder the terms: -0.75 + y = 1.060660172 Solving -0.75 + y = 1.060660172 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + y = 1.060660172 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + y = 1.060660172 + 0.75 y = 1.060660172 + 0.75 Combine like terms: 1.060660172 + 0.75 = 1.810660172 y = 1.810660172 Simplifying y = 1.810660172## Subproblem 2

y + -0.75 = -1.060660172 Simplifying y + -0.75 = -1.060660172 Reorder the terms: -0.75 + y = -1.060660172 Solving -0.75 + y = -1.060660172 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + y = -1.060660172 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + y = -1.060660172 + 0.75 y = -1.060660172 + 0.75 Combine like terms: -1.060660172 + 0.75 = -0.310660172 y = -0.310660172 Simplifying y = -0.310660172## Solution

The solution to the problem is based on the solutions from the subproblems. y = {1.810660172, -0.310660172}

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