y^2-20y+77=0

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Solution for y^2-20y+77=0 equation:


Simplifying
y2 + -20y + 77 = 0

Reorder the terms:
77 + -20y + y2 = 0

Solving
77 + -20y + y2 = 0

Solving for variable 'y'.

Begin completing the square.

Move the constant term to the right:

Add '-77' to each side of the equation.
77 + -20y + -77 + y2 = 0 + -77

Reorder the terms:
77 + -77 + -20y + y2 = 0 + -77

Combine like terms: 77 + -77 = 0
0 + -20y + y2 = 0 + -77
-20y + y2 = 0 + -77

Combine like terms: 0 + -77 = -77
-20y + y2 = -77

The y term is -20y.  Take half its coefficient (-10).
Square it (100) and add it to both sides.

Add '100' to each side of the equation.
-20y + 100 + y2 = -77 + 100

Reorder the terms:
100 + -20y + y2 = -77 + 100

Combine like terms: -77 + 100 = 23
100 + -20y + y2 = 23

Factor a perfect square on the left side:
(y + -10)(y + -10) = 23

Calculate the square root of the right side: 4.795831523

Break this problem into two subproblems by setting 
(y + -10) equal to 4.795831523 and -4.795831523.

Subproblem 1

y + -10 = 4.795831523 Simplifying y + -10 = 4.795831523 Reorder the terms: -10 + y = 4.795831523 Solving -10 + y = 4.795831523 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '10' to each side of the equation. -10 + 10 + y = 4.795831523 + 10 Combine like terms: -10 + 10 = 0 0 + y = 4.795831523 + 10 y = 4.795831523 + 10 Combine like terms: 4.795831523 + 10 = 14.795831523 y = 14.795831523 Simplifying y = 14.795831523

Subproblem 2

y + -10 = -4.795831523 Simplifying y + -10 = -4.795831523 Reorder the terms: -10 + y = -4.795831523 Solving -10 + y = -4.795831523 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '10' to each side of the equation. -10 + 10 + y = -4.795831523 + 10 Combine like terms: -10 + 10 = 0 0 + y = -4.795831523 + 10 y = -4.795831523 + 10 Combine like terms: -4.795831523 + 10 = 5.204168477 y = 5.204168477 Simplifying y = 5.204168477

Solution

The solution to the problem is based on the solutions from the subproblems. y = {14.795831523, 5.204168477}

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