4y-3y(y-3)+3=5

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Solution for 4y-3y(y-3)+3=5 equation:


Simplifying
4y + -3y(y + -3) + 3 = 5

Reorder the terms:
4y + -3y(-3 + y) + 3 = 5
4y + (-3 * -3y + y * -3y) + 3 = 5
4y + (9y + -3y2) + 3 = 5

Reorder the terms:
3 + 4y + 9y + -3y2 = 5

Combine like terms: 4y + 9y = 13y
3 + 13y + -3y2 = 5

Solving
3 + 13y + -3y2 = 5

Solving for variable 'y'.

Reorder the terms:
3 + -5 + 13y + -3y2 = 5 + -5

Combine like terms: 3 + -5 = -2
-2 + 13y + -3y2 = 5 + -5

Combine like terms: 5 + -5 = 0
-2 + 13y + -3y2 = 0

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

Divide each side by '-3'.
0.6666666667 + -4.333333333y + y2 = 0

Move the constant term to the right:

Add '-0.6666666667' to each side of the equation.
0.6666666667 + -4.333333333y + -0.6666666667 + y2 = 0 + -0.6666666667

Reorder the terms:
0.6666666667 + -0.6666666667 + -4.333333333y + y2 = 0 + -0.6666666667

Combine like terms: 0.6666666667 + -0.6666666667 = 0.0000000000
0.0000000000 + -4.333333333y + y2 = 0 + -0.6666666667
-4.333333333y + y2 = 0 + -0.6666666667

Combine like terms: 0 + -0.6666666667 = -0.6666666667
-4.333333333y + y2 = -0.6666666667

The y term is -4.333333333y.  Take half its coefficient (-2.166666667).
Square it (4.694444446) and add it to both sides.

Add '4.694444446' to each side of the equation.
-4.333333333y + 4.694444446 + y2 = -0.6666666667 + 4.694444446

Reorder the terms:
4.694444446 + -4.333333333y + y2 = -0.6666666667 + 4.694444446

Combine like terms: -0.6666666667 + 4.694444446 = 4.0277777793
4.694444446 + -4.333333333y + y2 = 4.0277777793

Factor a perfect square on the left side:
(y + -2.166666667)(y + -2.166666667) = 4.0277777793

Calculate the square root of the right side: 2.00693243

Break this problem into two subproblems by setting 
(y + -2.166666667) equal to 2.00693243 and -2.00693243.

Subproblem 1

y + -2.166666667 = 2.00693243 Simplifying y + -2.166666667 = 2.00693243 Reorder the terms: -2.166666667 + y = 2.00693243 Solving -2.166666667 + y = 2.00693243 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '2.166666667' to each side of the equation. -2.166666667 + 2.166666667 + y = 2.00693243 + 2.166666667 Combine like terms: -2.166666667 + 2.166666667 = 0.000000000 0.000000000 + y = 2.00693243 + 2.166666667 y = 2.00693243 + 2.166666667 Combine like terms: 2.00693243 + 2.166666667 = 4.173599097 y = 4.173599097 Simplifying y = 4.173599097

Subproblem 2

y + -2.166666667 = -2.00693243 Simplifying y + -2.166666667 = -2.00693243 Reorder the terms: -2.166666667 + y = -2.00693243 Solving -2.166666667 + y = -2.00693243 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '2.166666667' to each side of the equation. -2.166666667 + 2.166666667 + y = -2.00693243 + 2.166666667 Combine like terms: -2.166666667 + 2.166666667 = 0.000000000 0.000000000 + y = -2.00693243 + 2.166666667 y = -2.00693243 + 2.166666667 Combine like terms: -2.00693243 + 2.166666667 = 0.159734237 y = 0.159734237 Simplifying y = 0.159734237

Solution

The solution to the problem is based on the solutions from the subproblems. y = {4.173599097, 0.159734237}

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