4y^2=14y+1

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Solution for 4y^2=14y+1 equation:


Simplifying
4y2 = 14y + 1

Reorder the terms:
4y2 = 1 + 14y

Solving
4y2 = 1 + 14y

Solving for variable 'y'.

Reorder the terms:
-1 + -14y + 4y2 = 1 + 14y + -1 + -14y

Reorder the terms:
-1 + -14y + 4y2 = 1 + -1 + 14y + -14y

Combine like terms: 1 + -1 = 0
-1 + -14y + 4y2 = 0 + 14y + -14y
-1 + -14y + 4y2 = 14y + -14y

Combine like terms: 14y + -14y = 0
-1 + -14y + 4y2 = 0

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

Divide each side by '4'.
-0.25 + -3.5y + y2 = 0

Move the constant term to the right:

Add '0.25' to each side of the equation.
-0.25 + -3.5y + 0.25 + y2 = 0 + 0.25

Reorder the terms:
-0.25 + 0.25 + -3.5y + y2 = 0 + 0.25

Combine like terms: -0.25 + 0.25 = 0.00
0.00 + -3.5y + y2 = 0 + 0.25
-3.5y + y2 = 0 + 0.25

Combine like terms: 0 + 0.25 = 0.25
-3.5y + y2 = 0.25

The y term is -3.5y.  Take half its coefficient (-1.75).
Square it (3.0625) and add it to both sides.

Add '3.0625' to each side of the equation.
-3.5y + 3.0625 + y2 = 0.25 + 3.0625

Reorder the terms:
3.0625 + -3.5y + y2 = 0.25 + 3.0625

Combine like terms: 0.25 + 3.0625 = 3.3125
3.0625 + -3.5y + y2 = 3.3125

Factor a perfect square on the left side:
(y + -1.75)(y + -1.75) = 3.3125

Calculate the square root of the right side: 1.820027472

Break this problem into two subproblems by setting 
(y + -1.75) equal to 1.820027472 and -1.820027472.

Subproblem 1

y + -1.75 = 1.820027472 Simplifying y + -1.75 = 1.820027472 Reorder the terms: -1.75 + y = 1.820027472 Solving -1.75 + y = 1.820027472 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.75' to each side of the equation. -1.75 + 1.75 + y = 1.820027472 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + y = 1.820027472 + 1.75 y = 1.820027472 + 1.75 Combine like terms: 1.820027472 + 1.75 = 3.570027472 y = 3.570027472 Simplifying y = 3.570027472

Subproblem 2

y + -1.75 = -1.820027472 Simplifying y + -1.75 = -1.820027472 Reorder the terms: -1.75 + y = -1.820027472 Solving -1.75 + y = -1.820027472 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.75' to each side of the equation. -1.75 + 1.75 + y = -1.820027472 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + y = -1.820027472 + 1.75 y = -1.820027472 + 1.75 Combine like terms: -1.820027472 + 1.75 = -0.070027472 y = -0.070027472 Simplifying y = -0.070027472

Solution

The solution to the problem is based on the solutions from the subproblems. y = {3.570027472, -0.070027472}

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