3y^2+7y=7

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Solution for 3y^2+7y=7 equation:


Simplifying
3y2 + 7y = 7

Reorder the terms:
7y + 3y2 = 7

Solving
7y + 3y2 = 7

Solving for variable 'y'.

Reorder the terms:
-7 + 7y + 3y2 = 7 + -7

Combine like terms: 7 + -7 = 0
-7 + 7y + 3y2 = 0

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

Divide each side by '3'.
-2.333333333 + 2.333333333y + y2 = 0

Move the constant term to the right:

Add '2.333333333' to each side of the equation.
-2.333333333 + 2.333333333y + 2.333333333 + y2 = 0 + 2.333333333

Reorder the terms:
-2.333333333 + 2.333333333 + 2.333333333y + y2 = 0 + 2.333333333

Combine like terms: -2.333333333 + 2.333333333 = 0.000000000
0.000000000 + 2.333333333y + y2 = 0 + 2.333333333
2.333333333y + y2 = 0 + 2.333333333

Combine like terms: 0 + 2.333333333 = 2.333333333
2.333333333y + y2 = 2.333333333

The y term is 2.333333333y.  Take half its coefficient (1.166666667).
Square it (1.361111112) and add it to both sides.

Add '1.361111112' to each side of the equation.
2.333333333y + 1.361111112 + y2 = 2.333333333 + 1.361111112

Reorder the terms:
1.361111112 + 2.333333333y + y2 = 2.333333333 + 1.361111112

Combine like terms: 2.333333333 + 1.361111112 = 3.694444445
1.361111112 + 2.333333333y + y2 = 3.694444445

Factor a perfect square on the left side:
(y + 1.166666667)(y + 1.166666667) = 3.694444445

Calculate the square root of the right side: 1.922093766

Break this problem into two subproblems by setting 
(y + 1.166666667) equal to 1.922093766 and -1.922093766.

Subproblem 1

y + 1.166666667 = 1.922093766 Simplifying y + 1.166666667 = 1.922093766 Reorder the terms: 1.166666667 + y = 1.922093766 Solving 1.166666667 + y = 1.922093766 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.166666667' to each side of the equation. 1.166666667 + -1.166666667 + y = 1.922093766 + -1.166666667 Combine like terms: 1.166666667 + -1.166666667 = 0.000000000 0.000000000 + y = 1.922093766 + -1.166666667 y = 1.922093766 + -1.166666667 Combine like terms: 1.922093766 + -1.166666667 = 0.755427099 y = 0.755427099 Simplifying y = 0.755427099

Subproblem 2

y + 1.166666667 = -1.922093766 Simplifying y + 1.166666667 = -1.922093766 Reorder the terms: 1.166666667 + y = -1.922093766 Solving 1.166666667 + y = -1.922093766 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.166666667' to each side of the equation. 1.166666667 + -1.166666667 + y = -1.922093766 + -1.166666667 Combine like terms: 1.166666667 + -1.166666667 = 0.000000000 0.000000000 + y = -1.922093766 + -1.166666667 y = -1.922093766 + -1.166666667 Combine like terms: -1.922093766 + -1.166666667 = -3.088760433 y = -3.088760433 Simplifying y = -3.088760433

Solution

The solution to the problem is based on the solutions from the subproblems. y = {0.755427099, -3.088760433}

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