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Simplifying 8y2 + 24y + 9 = 0 Reorder the terms: 9 + 24y + 8y2 = 0 Solving 9 + 24y + 8y2 = 0 Solving for variable 'y'. Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. 1.125 + 3y + y2 = 0 Move the constant term to the right: Add '-1.125' to each side of the equation. 1.125 + 3y + -1.125 + y2 = 0 + -1.125 Reorder the terms: 1.125 + -1.125 + 3y + y2 = 0 + -1.125 Combine like terms: 1.125 + -1.125 = 0.000 0.000 + 3y + y2 = 0 + -1.125 3y + y2 = 0 + -1.125 Combine like terms: 0 + -1.125 = -1.125 3y + y2 = -1.125 The y term is 3y. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3y + 2.25 + y2 = -1.125 + 2.25 Reorder the terms: 2.25 + 3y + y2 = -1.125 + 2.25 Combine like terms: -1.125 + 2.25 = 1.125 2.25 + 3y + y2 = 1.125 Factor a perfect square on the left side: (y + 1.5)(y + 1.5) = 1.125 Calculate the square root of the right side: 1.060660172 Break this problem into two subproblems by setting (y + 1.5) equal to 1.060660172 and -1.060660172.Subproblem 1
y + 1.5 = 1.060660172 Simplifying y + 1.5 = 1.060660172 Reorder the terms: 1.5 + y = 1.060660172 Solving 1.5 + y = 1.060660172 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + y = 1.060660172 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + y = 1.060660172 + -1.5 y = 1.060660172 + -1.5 Combine like terms: 1.060660172 + -1.5 = -0.439339828 y = -0.439339828 Simplifying y = -0.439339828Subproblem 2
y + 1.5 = -1.060660172 Simplifying y + 1.5 = -1.060660172 Reorder the terms: 1.5 + y = -1.060660172 Solving 1.5 + y = -1.060660172 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + y = -1.060660172 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + y = -1.060660172 + -1.5 y = -1.060660172 + -1.5 Combine like terms: -1.060660172 + -1.5 = -2.560660172 y = -2.560660172 Simplifying y = -2.560660172Solution
The solution to the problem is based on the solutions from the subproblems. y = {-0.439339828, -2.560660172}
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