5y^2=7y+5

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Solution for 5y^2=7y+5 equation:


Simplifying
5y2 = 7y + 5

Reorder the terms:
5y2 = 5 + 7y

Solving
5y2 = 5 + 7y

Solving for variable 'y'.

Reorder the terms:
-5 + -7y + 5y2 = 5 + 7y + -5 + -7y

Reorder the terms:
-5 + -7y + 5y2 = 5 + -5 + 7y + -7y

Combine like terms: 5 + -5 = 0
-5 + -7y + 5y2 = 0 + 7y + -7y
-5 + -7y + 5y2 = 7y + -7y

Combine like terms: 7y + -7y = 0
-5 + -7y + 5y2 = 0

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

Divide each side by '5'.
-1 + -1.4y + y2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -1.4y + 1 + y2 = 0 + 1

Reorder the terms:
-1 + 1 + -1.4y + y2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + -1.4y + y2 = 0 + 1
-1.4y + y2 = 0 + 1

Combine like terms: 0 + 1 = 1
-1.4y + y2 = 1

The y term is -1.4y.  Take half its coefficient (-0.7).
Square it (0.49) and add it to both sides.

Add '0.49' to each side of the equation.
-1.4y + 0.49 + y2 = 1 + 0.49

Reorder the terms:
0.49 + -1.4y + y2 = 1 + 0.49

Combine like terms: 1 + 0.49 = 1.49
0.49 + -1.4y + y2 = 1.49

Factor a perfect square on the left side:
(y + -0.7)(y + -0.7) = 1.49

Calculate the square root of the right side: 1.220655562

Break this problem into two subproblems by setting 
(y + -0.7) equal to 1.220655562 and -1.220655562.

Subproblem 1

y + -0.7 = 1.220655562 Simplifying y + -0.7 = 1.220655562 Reorder the terms: -0.7 + y = 1.220655562 Solving -0.7 + y = 1.220655562 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.7' to each side of the equation. -0.7 + 0.7 + y = 1.220655562 + 0.7 Combine like terms: -0.7 + 0.7 = 0.0 0.0 + y = 1.220655562 + 0.7 y = 1.220655562 + 0.7 Combine like terms: 1.220655562 + 0.7 = 1.920655562 y = 1.920655562 Simplifying y = 1.920655562

Subproblem 2

y + -0.7 = -1.220655562 Simplifying y + -0.7 = -1.220655562 Reorder the terms: -0.7 + y = -1.220655562 Solving -0.7 + y = -1.220655562 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.7' to each side of the equation. -0.7 + 0.7 + y = -1.220655562 + 0.7 Combine like terms: -0.7 + 0.7 = 0.0 0.0 + y = -1.220655562 + 0.7 y = -1.220655562 + 0.7 Combine like terms: -1.220655562 + 0.7 = -0.520655562 y = -0.520655562 Simplifying y = -0.520655562

Solution

The solution to the problem is based on the solutions from the subproblems. y = {1.920655562, -0.520655562}

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