7y^2=8y+1

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Solution for 7y^2=8y+1 equation:


Simplifying
7y2 = 8y + 1

Reorder the terms:
7y2 = 1 + 8y

Solving
7y2 = 1 + 8y

Solving for variable 'y'.

Reorder the terms:
-1 + -8y + 7y2 = 1 + 8y + -1 + -8y

Reorder the terms:
-1 + -8y + 7y2 = 1 + -1 + 8y + -8y

Combine like terms: 1 + -1 = 0
-1 + -8y + 7y2 = 0 + 8y + -8y
-1 + -8y + 7y2 = 8y + -8y

Combine like terms: 8y + -8y = 0
-1 + -8y + 7y2 = 0

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

Divide each side by '7'.
-0.1428571429 + -1.142857143y + y2 = 0

Move the constant term to the right:

Add '0.1428571429' to each side of the equation.
-0.1428571429 + -1.142857143y + 0.1428571429 + y2 = 0 + 0.1428571429

Reorder the terms:
-0.1428571429 + 0.1428571429 + -1.142857143y + y2 = 0 + 0.1428571429

Combine like terms: -0.1428571429 + 0.1428571429 = 0.0000000000
0.0000000000 + -1.142857143y + y2 = 0 + 0.1428571429
-1.142857143y + y2 = 0 + 0.1428571429

Combine like terms: 0 + 0.1428571429 = 0.1428571429
-1.142857143y + y2 = 0.1428571429

The y term is -1.142857143y.  Take half its coefficient (-0.5714285715).
Square it (0.3265306123) and add it to both sides.

Add '0.3265306123' to each side of the equation.
-1.142857143y + 0.3265306123 + y2 = 0.1428571429 + 0.3265306123

Reorder the terms:
0.3265306123 + -1.142857143y + y2 = 0.1428571429 + 0.3265306123

Combine like terms: 0.1428571429 + 0.3265306123 = 0.4693877552
0.3265306123 + -1.142857143y + y2 = 0.4693877552

Factor a perfect square on the left side:
(y + -0.5714285715)(y + -0.5714285715) = 0.4693877552

Calculate the square root of the right side: 0.685118789

Break this problem into two subproblems by setting 
(y + -0.5714285715) equal to 0.685118789 and -0.685118789.

Subproblem 1

y + -0.5714285715 = 0.685118789 Simplifying y + -0.5714285715 = 0.685118789 Reorder the terms: -0.5714285715 + y = 0.685118789 Solving -0.5714285715 + y = 0.685118789 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.5714285715' to each side of the equation. -0.5714285715 + 0.5714285715 + y = 0.685118789 + 0.5714285715 Combine like terms: -0.5714285715 + 0.5714285715 = 0.0000000000 0.0000000000 + y = 0.685118789 + 0.5714285715 y = 0.685118789 + 0.5714285715 Combine like terms: 0.685118789 + 0.5714285715 = 1.2565473605 y = 1.2565473605 Simplifying y = 1.2565473605

Subproblem 2

y + -0.5714285715 = -0.685118789 Simplifying y + -0.5714285715 = -0.685118789 Reorder the terms: -0.5714285715 + y = -0.685118789 Solving -0.5714285715 + y = -0.685118789 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.5714285715' to each side of the equation. -0.5714285715 + 0.5714285715 + y = -0.685118789 + 0.5714285715 Combine like terms: -0.5714285715 + 0.5714285715 = 0.0000000000 0.0000000000 + y = -0.685118789 + 0.5714285715 y = -0.685118789 + 0.5714285715 Combine like terms: -0.685118789 + 0.5714285715 = -0.1136902175 y = -0.1136902175 Simplifying y = -0.1136902175

Solution

The solution to the problem is based on the solutions from the subproblems. y = {1.2565473605, -0.1136902175}

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