10y+20=y^2+4y

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Solution for 10y+20=y^2+4y equation:


Simplifying
10y + 20 = y2 + 4y

Reorder the terms:
20 + 10y = y2 + 4y

Reorder the terms:
20 + 10y = 4y + y2

Solving
20 + 10y = 4y + y2

Solving for variable 'y'.

Combine like terms: 10y + -4y = 6y
20 + 6y + -1y2 = 4y + y2 + -4y + -1y2

Reorder the terms:
20 + 6y + -1y2 = 4y + -4y + y2 + -1y2

Combine like terms: 4y + -4y = 0
20 + 6y + -1y2 = 0 + y2 + -1y2
20 + 6y + -1y2 = y2 + -1y2

Combine like terms: y2 + -1y2 = 0
20 + 6y + -1y2 = 0

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

Divide each side by '-1'.
-20 + -6y + y2 = 0

Move the constant term to the right:

Add '20' to each side of the equation.
-20 + -6y + 20 + y2 = 0 + 20

Reorder the terms:
-20 + 20 + -6y + y2 = 0 + 20

Combine like terms: -20 + 20 = 0
0 + -6y + y2 = 0 + 20
-6y + y2 = 0 + 20

Combine like terms: 0 + 20 = 20
-6y + y2 = 20

The y term is -6y.  Take half its coefficient (-3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
-6y + 9 + y2 = 20 + 9

Reorder the terms:
9 + -6y + y2 = 20 + 9

Combine like terms: 20 + 9 = 29
9 + -6y + y2 = 29

Factor a perfect square on the left side:
(y + -3)(y + -3) = 29

Calculate the square root of the right side: 5.385164807

Break this problem into two subproblems by setting 
(y + -3) equal to 5.385164807 and -5.385164807.

Subproblem 1

y + -3 = 5.385164807 Simplifying y + -3 = 5.385164807 Reorder the terms: -3 + y = 5.385164807 Solving -3 + y = 5.385164807 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + y = 5.385164807 + 3 Combine like terms: -3 + 3 = 0 0 + y = 5.385164807 + 3 y = 5.385164807 + 3 Combine like terms: 5.385164807 + 3 = 8.385164807 y = 8.385164807 Simplifying y = 8.385164807

Subproblem 2

y + -3 = -5.385164807 Simplifying y + -3 = -5.385164807 Reorder the terms: -3 + y = -5.385164807 Solving -3 + y = -5.385164807 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + y = -5.385164807 + 3 Combine like terms: -3 + 3 = 0 0 + y = -5.385164807 + 3 y = -5.385164807 + 3 Combine like terms: -5.385164807 + 3 = -2.385164807 y = -2.385164807 Simplifying y = -2.385164807

Solution

The solution to the problem is based on the solutions from the subproblems. y = {8.385164807, -2.385164807}

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