0=y^2-21y+66

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Solution for 0=y^2-21y+66 equation:


Simplifying
0 = y2 + -21y + 66

Reorder the terms:
0 = 66 + -21y + y2

Solving
0 = 66 + -21y + y2

Solving for variable 'y'.

Combine like terms: 0 + -66 = -66
-66 + 21y + -1y2 = 66 + -21y + y2 + -66 + 21y + -1y2

Reorder the terms:
-66 + 21y + -1y2 = 66 + -66 + -21y + 21y + y2 + -1y2

Combine like terms: 66 + -66 = 0
-66 + 21y + -1y2 = 0 + -21y + 21y + y2 + -1y2
-66 + 21y + -1y2 = -21y + 21y + y2 + -1y2

Combine like terms: -21y + 21y = 0
-66 + 21y + -1y2 = 0 + y2 + -1y2
-66 + 21y + -1y2 = y2 + -1y2

Combine like terms: y2 + -1y2 = 0
-66 + 21y + -1y2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
66 + -21y + y2 = 0

Move the constant term to the right:

Add '-66' to each side of the equation.
66 + -21y + -66 + y2 = 0 + -66

Reorder the terms:
66 + -66 + -21y + y2 = 0 + -66

Combine like terms: 66 + -66 = 0
0 + -21y + y2 = 0 + -66
-21y + y2 = 0 + -66

Combine like terms: 0 + -66 = -66
-21y + y2 = -66

The y term is -21y.  Take half its coefficient (-10.5).
Square it (110.25) and add it to both sides.

Add '110.25' to each side of the equation.
-21y + 110.25 + y2 = -66 + 110.25

Reorder the terms:
110.25 + -21y + y2 = -66 + 110.25

Combine like terms: -66 + 110.25 = 44.25
110.25 + -21y + y2 = 44.25

Factor a perfect square on the left side:
(y + -10.5)(y + -10.5) = 44.25

Calculate the square root of the right side: 6.652067348

Break this problem into two subproblems by setting 
(y + -10.5) equal to 6.652067348 and -6.652067348.

Subproblem 1

y + -10.5 = 6.652067348 Simplifying y + -10.5 = 6.652067348 Reorder the terms: -10.5 + y = 6.652067348 Solving -10.5 + y = 6.652067348 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '10.5' to each side of the equation. -10.5 + 10.5 + y = 6.652067348 + 10.5 Combine like terms: -10.5 + 10.5 = 0.0 0.0 + y = 6.652067348 + 10.5 y = 6.652067348 + 10.5 Combine like terms: 6.652067348 + 10.5 = 17.152067348 y = 17.152067348 Simplifying y = 17.152067348

Subproblem 2

y + -10.5 = -6.652067348 Simplifying y + -10.5 = -6.652067348 Reorder the terms: -10.5 + y = -6.652067348 Solving -10.5 + y = -6.652067348 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '10.5' to each side of the equation. -10.5 + 10.5 + y = -6.652067348 + 10.5 Combine like terms: -10.5 + 10.5 = 0.0 0.0 + y = -6.652067348 + 10.5 y = -6.652067348 + 10.5 Combine like terms: -6.652067348 + 10.5 = 3.847932652 y = 3.847932652 Simplifying y = 3.847932652

Solution

The solution to the problem is based on the solutions from the subproblems. y = {17.152067348, 3.847932652}

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