4y^2+21y=5

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Solution for 4y^2+21y=5 equation:


Simplifying
4y2 + 21y = 5

Reorder the terms:
21y + 4y2 = 5

Solving
21y + 4y2 = 5

Solving for variable 'y'.

Reorder the terms:
-5 + 21y + 4y2 = 5 + -5

Combine like terms: 5 + -5 = 0
-5 + 21y + 4y2 = 0

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

Divide each side by '4'.
-1.25 + 5.25y + y2 = 0

Move the constant term to the right:

Add '1.25' to each side of the equation.
-1.25 + 5.25y + 1.25 + y2 = 0 + 1.25

Reorder the terms:
-1.25 + 1.25 + 5.25y + y2 = 0 + 1.25

Combine like terms: -1.25 + 1.25 = 0.00
0.00 + 5.25y + y2 = 0 + 1.25
5.25y + y2 = 0 + 1.25

Combine like terms: 0 + 1.25 = 1.25
5.25y + y2 = 1.25

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 = 1.25 + 6.890625

Reorder the terms:
6.890625 + 5.25y + y2 = 1.25 + 6.890625

Combine like terms: 1.25 + 6.890625 = 8.140625
6.890625 + 5.25y + y2 = 8.140625

Factor a perfect square on the left side:
(y + 2.625)(y + 2.625) = 8.140625

Calculate the square root of the right side: 2.853178053

Break this problem into two subproblems by setting 
(y + 2.625) equal to 2.853178053 and -2.853178053.

Subproblem 1

y + 2.625 = 2.853178053 Simplifying y + 2.625 = 2.853178053 Reorder the terms: 2.625 + y = 2.853178053 Solving 2.625 + y = 2.853178053 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 = 2.853178053 + -2.625 Combine like terms: 2.625 + -2.625 = 0.000 0.000 + y = 2.853178053 + -2.625 y = 2.853178053 + -2.625 Combine like terms: 2.853178053 + -2.625 = 0.228178053 y = 0.228178053 Simplifying y = 0.228178053

Subproblem 2

y + 2.625 = -2.853178053 Simplifying y + 2.625 = -2.853178053 Reorder the terms: 2.625 + y = -2.853178053 Solving 2.625 + y = -2.853178053 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 = -2.853178053 + -2.625 Combine like terms: 2.625 + -2.625 = 0.000 0.000 + y = -2.853178053 + -2.625 y = -2.853178053 + -2.625 Combine like terms: -2.853178053 + -2.625 = -5.478178053 y = -5.478178053 Simplifying y = -5.478178053

Solution

The solution to the problem is based on the solutions from the subproblems. y = {0.228178053, -5.478178053}

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