7(Y+7)Y=6

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Solution for 7(Y+7)Y=6 equation:


Simplifying
7(Y + 7) * Y = 6

Reorder the terms:
7(7 + Y) * Y = 6

Reorder the terms for easier multiplication:
7Y(7 + Y) = 6
(7 * 7Y + Y * 7Y) = 6
(49Y + 7Y2) = 6

Solving
49Y + 7Y2 = 6

Solving for variable 'Y'.

Reorder the terms:
-6 + 49Y + 7Y2 = 6 + -6

Combine like terms: 6 + -6 = 0
-6 + 49Y + 7Y2 = 0

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

Divide each side by '7'.
-0.8571428571 + 7Y + Y2 = 0

Move the constant term to the right:

Add '0.8571428571' to each side of the equation.
-0.8571428571 + 7Y + 0.8571428571 + Y2 = 0 + 0.8571428571

Reorder the terms:
-0.8571428571 + 0.8571428571 + 7Y + Y2 = 0 + 0.8571428571

Combine like terms: -0.8571428571 + 0.8571428571 = 0.0000000000
0.0000000000 + 7Y + Y2 = 0 + 0.8571428571
7Y + Y2 = 0 + 0.8571428571

Combine like terms: 0 + 0.8571428571 = 0.8571428571
7Y + Y2 = 0.8571428571

The Y term is 7Y.  Take half its coefficient (3.5).
Square it (12.25) and add it to both sides.

Add '12.25' to each side of the equation.
7Y + 12.25 + Y2 = 0.8571428571 + 12.25

Reorder the terms:
12.25 + 7Y + Y2 = 0.8571428571 + 12.25

Combine like terms: 0.8571428571 + 12.25 = 13.1071428571
12.25 + 7Y + Y2 = 13.1071428571

Factor a perfect square on the left side:
(Y + 3.5)(Y + 3.5) = 13.1071428571

Calculate the square root of the right side: 3.620378828

Break this problem into two subproblems by setting 
(Y + 3.5) equal to 3.620378828 and -3.620378828.

Subproblem 1

Y + 3.5 = 3.620378828 Simplifying Y + 3.5 = 3.620378828 Reorder the terms: 3.5 + Y = 3.620378828 Solving 3.5 + Y = 3.620378828 Solving for variable 'Y'. Move all terms containing Y to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + Y = 3.620378828 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + Y = 3.620378828 + -3.5 Y = 3.620378828 + -3.5 Combine like terms: 3.620378828 + -3.5 = 0.120378828 Y = 0.120378828 Simplifying Y = 0.120378828

Subproblem 2

Y + 3.5 = -3.620378828 Simplifying Y + 3.5 = -3.620378828 Reorder the terms: 3.5 + Y = -3.620378828 Solving 3.5 + Y = -3.620378828 Solving for variable 'Y'. Move all terms containing Y to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + Y = -3.620378828 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + Y = -3.620378828 + -3.5 Y = -3.620378828 + -3.5 Combine like terms: -3.620378828 + -3.5 = -7.120378828 Y = -7.120378828 Simplifying Y = -7.120378828

Solution

The solution to the problem is based on the solutions from the subproblems. Y = {0.120378828, -7.120378828}

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