y^2+7y=4

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Solution for y^2+7y=4 equation:


Simplifying
y2 + 7y = 4

Reorder the terms:
7y + y2 = 4

Solving
7y + y2 = 4

Solving for variable 'y'.

Reorder the terms:
-4 + 7y + y2 = 4 + -4

Combine like terms: 4 + -4 = 0
-4 + 7y + y2 = 0

Begin completing the square.

Move the constant term to the right:

Add '4' to each side of the equation.
-4 + 7y + 4 + y2 = 0 + 4

Reorder the terms:
-4 + 4 + 7y + y2 = 0 + 4

Combine like terms: -4 + 4 = 0
0 + 7y + y2 = 0 + 4
7y + y2 = 0 + 4

Combine like terms: 0 + 4 = 4
7y + y2 = 4

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 = 4 + 12.25

Reorder the terms:
12.25 + 7y + y2 = 4 + 12.25

Combine like terms: 4 + 12.25 = 16.25
12.25 + 7y + y2 = 16.25

Factor a perfect square on the left side:
(y + 3.5)(y + 3.5) = 16.25

Calculate the square root of the right side: 4.031128874

Break this problem into two subproblems by setting 
(y + 3.5) equal to 4.031128874 and -4.031128874.

Subproblem 1

y + 3.5 = 4.031128874 Simplifying y + 3.5 = 4.031128874 Reorder the terms: 3.5 + y = 4.031128874 Solving 3.5 + y = 4.031128874 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 = 4.031128874 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + y = 4.031128874 + -3.5 y = 4.031128874 + -3.5 Combine like terms: 4.031128874 + -3.5 = 0.531128874 y = 0.531128874 Simplifying y = 0.531128874

Subproblem 2

y + 3.5 = -4.031128874 Simplifying y + 3.5 = -4.031128874 Reorder the terms: 3.5 + y = -4.031128874 Solving 3.5 + y = -4.031128874 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 = -4.031128874 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + y = -4.031128874 + -3.5 y = -4.031128874 + -3.5 Combine like terms: -4.031128874 + -3.5 = -7.531128874 y = -7.531128874 Simplifying y = -7.531128874

Solution

The solution to the problem is based on the solutions from the subproblems. y = {0.531128874, -7.531128874}

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