4y^2=3-5y

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Solution for 4y^2=3-5y equation:


Simplifying
4y2 = 3 + -5y

Solving
4y2 = 3 + -5y

Solving for variable 'y'.

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

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

Combine like terms: 3 + -3 = 0
-3 + 5y + 4y2 = 0 + -5y + 5y
-3 + 5y + 4y2 = -5y + 5y

Combine like terms: -5y + 5y = 0
-3 + 5y + 4y2 = 0

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

Divide each side by '4'.
-0.75 + 1.25y + y2 = 0

Move the constant term to the right:

Add '0.75' to each side of the equation.
-0.75 + 1.25y + 0.75 + y2 = 0 + 0.75

Reorder the terms:
-0.75 + 0.75 + 1.25y + y2 = 0 + 0.75

Combine like terms: -0.75 + 0.75 = 0.00
0.00 + 1.25y + y2 = 0 + 0.75
1.25y + y2 = 0 + 0.75

Combine like terms: 0 + 0.75 = 0.75
1.25y + y2 = 0.75

The y term is 1.25y.  Take half its coefficient (0.625).
Square it (0.390625) and add it to both sides.

Add '0.390625' to each side of the equation.
1.25y + 0.390625 + y2 = 0.75 + 0.390625

Reorder the terms:
0.390625 + 1.25y + y2 = 0.75 + 0.390625

Combine like terms: 0.75 + 0.390625 = 1.140625
0.390625 + 1.25y + y2 = 1.140625

Factor a perfect square on the left side:
(y + 0.625)(y + 0.625) = 1.140625

Calculate the square root of the right side: 1.068000468

Break this problem into two subproblems by setting 
(y + 0.625) equal to 1.068000468 and -1.068000468.

Subproblem 1

y + 0.625 = 1.068000468 Simplifying y + 0.625 = 1.068000468 Reorder the terms: 0.625 + y = 1.068000468 Solving 0.625 + y = 1.068000468 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.625' to each side of the equation. 0.625 + -0.625 + y = 1.068000468 + -0.625 Combine like terms: 0.625 + -0.625 = 0.000 0.000 + y = 1.068000468 + -0.625 y = 1.068000468 + -0.625 Combine like terms: 1.068000468 + -0.625 = 0.443000468 y = 0.443000468 Simplifying y = 0.443000468

Subproblem 2

y + 0.625 = -1.068000468 Simplifying y + 0.625 = -1.068000468 Reorder the terms: 0.625 + y = -1.068000468 Solving 0.625 + y = -1.068000468 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.625' to each side of the equation. 0.625 + -0.625 + y = -1.068000468 + -0.625 Combine like terms: 0.625 + -0.625 = 0.000 0.000 + y = -1.068000468 + -0.625 y = -1.068000468 + -0.625 Combine like terms: -1.068000468 + -0.625 = -1.693000468 y = -1.693000468 Simplifying y = -1.693000468

Solution

The solution to the problem is based on the solutions from the subproblems. y = {0.443000468, -1.693000468}

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