3y+4=6y(4y+6)

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Solution for 3y+4=6y(4y+6) equation:


Simplifying
3y + 4 = 6y(4y + 6)

Reorder the terms:
4 + 3y = 6y(4y + 6)

Reorder the terms:
4 + 3y = 6y(6 + 4y)
4 + 3y = (6 * 6y + 4y * 6y)
4 + 3y = (36y + 24y2)

Solving
4 + 3y = 36y + 24y2

Solving for variable 'y'.

Combine like terms: 3y + -36y = -33y
4 + -33y + -24y2 = 36y + 24y2 + -36y + -24y2

Reorder the terms:
4 + -33y + -24y2 = 36y + -36y + 24y2 + -24y2

Combine like terms: 36y + -36y = 0
4 + -33y + -24y2 = 0 + 24y2 + -24y2
4 + -33y + -24y2 = 24y2 + -24y2

Combine like terms: 24y2 + -24y2 = 0
4 + -33y + -24y2 = 0

Begin completing the square.  Divide all terms by
-24 the coefficient of the squared term: 

Divide each side by '-24'.
-0.1666666667 + 1.375y + y2 = 0

Move the constant term to the right:

Add '0.1666666667' to each side of the equation.
-0.1666666667 + 1.375y + 0.1666666667 + y2 = 0 + 0.1666666667

Reorder the terms:
-0.1666666667 + 0.1666666667 + 1.375y + y2 = 0 + 0.1666666667

Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000
0.0000000000 + 1.375y + y2 = 0 + 0.1666666667
1.375y + y2 = 0 + 0.1666666667

Combine like terms: 0 + 0.1666666667 = 0.1666666667
1.375y + y2 = 0.1666666667

The y term is 1.375y.  Take half its coefficient (0.6875).
Square it (0.47265625) and add it to both sides.

Add '0.47265625' to each side of the equation.
1.375y + 0.47265625 + y2 = 0.1666666667 + 0.47265625

Reorder the terms:
0.47265625 + 1.375y + y2 = 0.1666666667 + 0.47265625

Combine like terms: 0.1666666667 + 0.47265625 = 0.6393229167
0.47265625 + 1.375y + y2 = 0.6393229167

Factor a perfect square on the left side:
(y + 0.6875)(y + 0.6875) = 0.6393229167

Calculate the square root of the right side: 0.799576711

Break this problem into two subproblems by setting 
(y + 0.6875) equal to 0.799576711 and -0.799576711.

Subproblem 1

y + 0.6875 = 0.799576711 Simplifying y + 0.6875 = 0.799576711 Reorder the terms: 0.6875 + y = 0.799576711 Solving 0.6875 + y = 0.799576711 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.6875' to each side of the equation. 0.6875 + -0.6875 + y = 0.799576711 + -0.6875 Combine like terms: 0.6875 + -0.6875 = 0.0000 0.0000 + y = 0.799576711 + -0.6875 y = 0.799576711 + -0.6875 Combine like terms: 0.799576711 + -0.6875 = 0.112076711 y = 0.112076711 Simplifying y = 0.112076711

Subproblem 2

y + 0.6875 = -0.799576711 Simplifying y + 0.6875 = -0.799576711 Reorder the terms: 0.6875 + y = -0.799576711 Solving 0.6875 + y = -0.799576711 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.6875' to each side of the equation. 0.6875 + -0.6875 + y = -0.799576711 + -0.6875 Combine like terms: 0.6875 + -0.6875 = 0.0000 0.0000 + y = -0.799576711 + -0.6875 y = -0.799576711 + -0.6875 Combine like terms: -0.799576711 + -0.6875 = -1.487076711 y = -1.487076711 Simplifying y = -1.487076711

Solution

The solution to the problem is based on the solutions from the subproblems. y = {0.112076711, -1.487076711}

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