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Simplifying y2 + 3y = 78 Reorder the terms: 3y + y2 = 78 Solving 3y + y2 = 78 Solving for variable 'y'. Reorder the terms: -78 + 3y + y2 = 78 + -78 Combine like terms: 78 + -78 = 0 -78 + 3y + y2 = 0 Begin completing the square. Move the constant term to the right: Add '78' to each side of the equation. -78 + 3y + 78 + y2 = 0 + 78 Reorder the terms: -78 + 78 + 3y + y2 = 0 + 78 Combine like terms: -78 + 78 = 0 0 + 3y + y2 = 0 + 78 3y + y2 = 0 + 78 Combine like terms: 0 + 78 = 78 3y + y2 = 78 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 = 78 + 2.25 Reorder the terms: 2.25 + 3y + y2 = 78 + 2.25 Combine like terms: 78 + 2.25 = 80.25 2.25 + 3y + y2 = 80.25 Factor a perfect square on the left side: (y + 1.5)(y + 1.5) = 80.25 Calculate the square root of the right side: 8.958236434 Break this problem into two subproblems by setting (y + 1.5) equal to 8.958236434 and -8.958236434.Subproblem 1
y + 1.5 = 8.958236434 Simplifying y + 1.5 = 8.958236434 Reorder the terms: 1.5 + y = 8.958236434 Solving 1.5 + y = 8.958236434 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 = 8.958236434 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + y = 8.958236434 + -1.5 y = 8.958236434 + -1.5 Combine like terms: 8.958236434 + -1.5 = 7.458236434 y = 7.458236434 Simplifying y = 7.458236434Subproblem 2
y + 1.5 = -8.958236434 Simplifying y + 1.5 = -8.958236434 Reorder the terms: 1.5 + y = -8.958236434 Solving 1.5 + y = -8.958236434 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 = -8.958236434 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + y = -8.958236434 + -1.5 y = -8.958236434 + -1.5 Combine like terms: -8.958236434 + -1.5 = -10.458236434 y = -10.458236434 Simplifying y = -10.458236434Solution
The solution to the problem is based on the solutions from the subproblems. y = {7.458236434, -10.458236434}
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