0=4y^2-21y+22

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Solution for 0=4y^2-21y+22 equation:


Simplifying
0 = 4y2 + -21y + 22

Reorder the terms:
0 = 22 + -21y + 4y2

Solving
0 = 22 + -21y + 4y2

Solving for variable 'y'.

Combine like terms: 0 + -22 = -22
-22 + 21y + -4y2 = 22 + -21y + 4y2 + -22 + 21y + -4y2

Reorder the terms:
-22 + 21y + -4y2 = 22 + -22 + -21y + 21y + 4y2 + -4y2

Combine like terms: 22 + -22 = 0
-22 + 21y + -4y2 = 0 + -21y + 21y + 4y2 + -4y2
-22 + 21y + -4y2 = -21y + 21y + 4y2 + -4y2

Combine like terms: -21y + 21y = 0
-22 + 21y + -4y2 = 0 + 4y2 + -4y2
-22 + 21y + -4y2 = 4y2 + -4y2

Combine like terms: 4y2 + -4y2 = 0
-22 + 21y + -4y2 = 0

Begin completing the square.  Divide all terms by
-4 the coefficient of the squared term: 

Divide each side by '-4'.
5.5 + -5.25y + y2 = 0

Move the constant term to the right:

Add '-5.5' to each side of the equation.
5.5 + -5.25y + -5.5 + y2 = 0 + -5.5

Reorder the terms:
5.5 + -5.5 + -5.25y + y2 = 0 + -5.5

Combine like terms: 5.5 + -5.5 = 0.0
0.0 + -5.25y + y2 = 0 + -5.5
-5.25y + y2 = 0 + -5.5

Combine like terms: 0 + -5.5 = -5.5
-5.25y + y2 = -5.5

The y term is -5.25y.  Take half its coefficient (-2.625).
Square it (6.890625) and add it to both sides.

Add '6.890625' to each side of the equation.
-5.25y + 6.890625 + y2 = -5.5 + 6.890625

Reorder the terms:
6.890625 + -5.25y + y2 = -5.5 + 6.890625

Combine like terms: -5.5 + 6.890625 = 1.390625
6.890625 + -5.25y + y2 = 1.390625

Factor a perfect square on the left side:
(y + -2.625)(y + -2.625) = 1.390625

Calculate the square root of the right side: 1.179247642

Break this problem into two subproblems by setting 
(y + -2.625) equal to 1.179247642 and -1.179247642.

Subproblem 1

y + -2.625 = 1.179247642 Simplifying y + -2.625 = 1.179247642 Reorder the terms: -2.625 + y = 1.179247642 Solving -2.625 + y = 1.179247642 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '2.625' to each side of the equation. -2.625 + 2.625 + y = 1.179247642 + 2.625 Combine like terms: -2.625 + 2.625 = 0.000 0.000 + y = 1.179247642 + 2.625 y = 1.179247642 + 2.625 Combine like terms: 1.179247642 + 2.625 = 3.804247642 y = 3.804247642 Simplifying y = 3.804247642

Subproblem 2

y + -2.625 = -1.179247642 Simplifying y + -2.625 = -1.179247642 Reorder the terms: -2.625 + y = -1.179247642 Solving -2.625 + y = -1.179247642 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '2.625' to each side of the equation. -2.625 + 2.625 + y = -1.179247642 + 2.625 Combine like terms: -2.625 + 2.625 = 0.000 0.000 + y = -1.179247642 + 2.625 y = -1.179247642 + 2.625 Combine like terms: -1.179247642 + 2.625 = 1.445752358 y = 1.445752358 Simplifying y = 1.445752358

Solution

The solution to the problem is based on the solutions from the subproblems. y = {3.804247642, 1.445752358}

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