3y^2+4=11y

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Solution for 3y^2+4=11y equation:


Simplifying
3y2 + 4 = 11y

Reorder the terms:
4 + 3y2 = 11y

Solving
4 + 3y2 = 11y

Solving for variable 'y'.

Reorder the terms:
4 + -11y + 3y2 = 11y + -11y

Combine like terms: 11y + -11y = 0
4 + -11y + 3y2 = 0

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

Divide each side by '3'.
1.333333333 + -3.666666667y + y2 = 0

Move the constant term to the right:

Add '-1.333333333' to each side of the equation.
1.333333333 + -3.666666667y + -1.333333333 + y2 = 0 + -1.333333333

Reorder the terms:
1.333333333 + -1.333333333 + -3.666666667y + y2 = 0 + -1.333333333

Combine like terms: 1.333333333 + -1.333333333 = 0.000000000
0.000000000 + -3.666666667y + y2 = 0 + -1.333333333
-3.666666667y + y2 = 0 + -1.333333333

Combine like terms: 0 + -1.333333333 = -1.333333333
-3.666666667y + y2 = -1.333333333

The y term is -3.666666667y.  Take half its coefficient (-1.833333334).
Square it (3.361111114) and add it to both sides.

Add '3.361111114' to each side of the equation.
-3.666666667y + 3.361111114 + y2 = -1.333333333 + 3.361111114

Reorder the terms:
3.361111114 + -3.666666667y + y2 = -1.333333333 + 3.361111114

Combine like terms: -1.333333333 + 3.361111114 = 2.027777781
3.361111114 + -3.666666667y + y2 = 2.027777781

Factor a perfect square on the left side:
(y + -1.833333334)(y + -1.833333334) = 2.027777781

Calculate the square root of the right side: 1.424000625

Break this problem into two subproblems by setting 
(y + -1.833333334) equal to 1.424000625 and -1.424000625.

Subproblem 1

y + -1.833333334 = 1.424000625 Simplifying y + -1.833333334 = 1.424000625 Reorder the terms: -1.833333334 + y = 1.424000625 Solving -1.833333334 + y = 1.424000625 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.833333334' to each side of the equation. -1.833333334 + 1.833333334 + y = 1.424000625 + 1.833333334 Combine like terms: -1.833333334 + 1.833333334 = 0.000000000 0.000000000 + y = 1.424000625 + 1.833333334 y = 1.424000625 + 1.833333334 Combine like terms: 1.424000625 + 1.833333334 = 3.257333959 y = 3.257333959 Simplifying y = 3.257333959

Subproblem 2

y + -1.833333334 = -1.424000625 Simplifying y + -1.833333334 = -1.424000625 Reorder the terms: -1.833333334 + y = -1.424000625 Solving -1.833333334 + y = -1.424000625 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.833333334' to each side of the equation. -1.833333334 + 1.833333334 + y = -1.424000625 + 1.833333334 Combine like terms: -1.833333334 + 1.833333334 = 0.000000000 0.000000000 + y = -1.424000625 + 1.833333334 y = -1.424000625 + 1.833333334 Combine like terms: -1.424000625 + 1.833333334 = 0.409332709 y = 0.409332709 Simplifying y = 0.409332709

Solution

The solution to the problem is based on the solutions from the subproblems. y = {3.257333959, 0.409332709}

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