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Simplifying y2 + 19y = 78 Reorder the terms: 19y + y2 = 78 Solving 19y + y2 = 78 Solving for variable 'y'. Reorder the terms: -78 + 19y + y2 = 78 + -78 Combine like terms: 78 + -78 = 0 -78 + 19y + y2 = 0 Begin completing the square. Move the constant term to the right: Add '78' to each side of the equation. -78 + 19y + 78 + y2 = 0 + 78 Reorder the terms: -78 + 78 + 19y + y2 = 0 + 78 Combine like terms: -78 + 78 = 0 0 + 19y + y2 = 0 + 78 19y + y2 = 0 + 78 Combine like terms: 0 + 78 = 78 19y + y2 = 78 The y term is 19y. Take half its coefficient (9.5). Square it (90.25) and add it to both sides. Add '90.25' to each side of the equation. 19y + 90.25 + y2 = 78 + 90.25 Reorder the terms: 90.25 + 19y + y2 = 78 + 90.25 Combine like terms: 78 + 90.25 = 168.25 90.25 + 19y + y2 = 168.25 Factor a perfect square on the left side: (y + 9.5)(y + 9.5) = 168.25 Calculate the square root of the right side: 12.971121771 Break this problem into two subproblems by setting (y + 9.5) equal to 12.971121771 and -12.971121771.Subproblem 1
y + 9.5 = 12.971121771 Simplifying y + 9.5 = 12.971121771 Reorder the terms: 9.5 + y = 12.971121771 Solving 9.5 + y = 12.971121771 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-9.5' to each side of the equation. 9.5 + -9.5 + y = 12.971121771 + -9.5 Combine like terms: 9.5 + -9.5 = 0.0 0.0 + y = 12.971121771 + -9.5 y = 12.971121771 + -9.5 Combine like terms: 12.971121771 + -9.5 = 3.471121771 y = 3.471121771 Simplifying y = 3.471121771Subproblem 2
y + 9.5 = -12.971121771 Simplifying y + 9.5 = -12.971121771 Reorder the terms: 9.5 + y = -12.971121771 Solving 9.5 + y = -12.971121771 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-9.5' to each side of the equation. 9.5 + -9.5 + y = -12.971121771 + -9.5 Combine like terms: 9.5 + -9.5 = 0.0 0.0 + y = -12.971121771 + -9.5 y = -12.971121771 + -9.5 Combine like terms: -12.971121771 + -9.5 = -22.471121771 y = -22.471121771 Simplifying y = -22.471121771Solution
The solution to the problem is based on the solutions from the subproblems. y = {3.471121771, -22.471121771}
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