4y^2+3y=4

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


Simplifying
4y2 + 3y = 4

Reorder the terms:
3y + 4y2 = 4

Solving
3y + 4y2 = 4

Solving for variable 'y'.

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

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

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

Divide each side by '4'.
-1 + 0.75y + y2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + 0.75y + 1 + y2 = 0 + 1

Reorder the terms:
-1 + 1 + 0.75y + y2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + 0.75y + y2 = 0 + 1
0.75y + y2 = 0 + 1

Combine like terms: 0 + 1 = 1
0.75y + y2 = 1

The y term is 0.75y.  Take half its coefficient (0.375).
Square it (0.140625) and add it to both sides.

Add '0.140625' to each side of the equation.
0.75y + 0.140625 + y2 = 1 + 0.140625

Reorder the terms:
0.140625 + 0.75y + y2 = 1 + 0.140625

Combine like terms: 1 + 0.140625 = 1.140625
0.140625 + 0.75y + y2 = 1.140625

Factor a perfect square on the left side:
(y + 0.375)(y + 0.375) = 1.140625

Calculate the square root of the right side: 1.068000468

Break this problem into two subproblems by setting 
(y + 0.375) equal to 1.068000468 and -1.068000468.

Subproblem 1

y + 0.375 = 1.068000468 Simplifying y + 0.375 = 1.068000468 Reorder the terms: 0.375 + y = 1.068000468 Solving 0.375 + y = 1.068000468 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.375' to each side of the equation. 0.375 + -0.375 + y = 1.068000468 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + y = 1.068000468 + -0.375 y = 1.068000468 + -0.375 Combine like terms: 1.068000468 + -0.375 = 0.693000468 y = 0.693000468 Simplifying y = 0.693000468

Subproblem 2

y + 0.375 = -1.068000468 Simplifying y + 0.375 = -1.068000468 Reorder the terms: 0.375 + y = -1.068000468 Solving 0.375 + y = -1.068000468 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.375' to each side of the equation. 0.375 + -0.375 + y = -1.068000468 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + y = -1.068000468 + -0.375 y = -1.068000468 + -0.375 Combine like terms: -1.068000468 + -0.375 = -1.443000468 y = -1.443000468 Simplifying y = -1.443000468

Solution

The solution to the problem is based on the solutions from the subproblems. y = {0.693000468, -1.443000468}

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