0=4y^2-15y+13

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Solution for 0=4y^2-15y+13 equation:


Simplifying
0 = 4y2 + -15y + 13

Reorder the terms:
0 = 13 + -15y + 4y2

Solving
0 = 13 + -15y + 4y2

Solving for variable 'y'.

Combine like terms: 0 + -13 = -13
-13 + 15y + -4y2 = 13 + -15y + 4y2 + -13 + 15y + -4y2

Reorder the terms:
-13 + 15y + -4y2 = 13 + -13 + -15y + 15y + 4y2 + -4y2

Combine like terms: 13 + -13 = 0
-13 + 15y + -4y2 = 0 + -15y + 15y + 4y2 + -4y2
-13 + 15y + -4y2 = -15y + 15y + 4y2 + -4y2

Combine like terms: -15y + 15y = 0
-13 + 15y + -4y2 = 0 + 4y2 + -4y2
-13 + 15y + -4y2 = 4y2 + -4y2

Combine like terms: 4y2 + -4y2 = 0
-13 + 15y + -4y2 = 0

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

Divide each side by '-4'.
3.25 + -3.75y + y2 = 0

Move the constant term to the right:

Add '-3.25' to each side of the equation.
3.25 + -3.75y + -3.25 + y2 = 0 + -3.25

Reorder the terms:
3.25 + -3.25 + -3.75y + y2 = 0 + -3.25

Combine like terms: 3.25 + -3.25 = 0.00
0.00 + -3.75y + y2 = 0 + -3.25
-3.75y + y2 = 0 + -3.25

Combine like terms: 0 + -3.25 = -3.25
-3.75y + y2 = -3.25

The y term is -3.75y.  Take half its coefficient (-1.875).
Square it (3.515625) and add it to both sides.

Add '3.515625' to each side of the equation.
-3.75y + 3.515625 + y2 = -3.25 + 3.515625

Reorder the terms:
3.515625 + -3.75y + y2 = -3.25 + 3.515625

Combine like terms: -3.25 + 3.515625 = 0.265625
3.515625 + -3.75y + y2 = 0.265625

Factor a perfect square on the left side:
(y + -1.875)(y + -1.875) = 0.265625

Calculate the square root of the right side: 0.515388203

Break this problem into two subproblems by setting 
(y + -1.875) equal to 0.515388203 and -0.515388203.

Subproblem 1

y + -1.875 = 0.515388203 Simplifying y + -1.875 = 0.515388203 Reorder the terms: -1.875 + y = 0.515388203 Solving -1.875 + y = 0.515388203 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.875' to each side of the equation. -1.875 + 1.875 + y = 0.515388203 + 1.875 Combine like terms: -1.875 + 1.875 = 0.000 0.000 + y = 0.515388203 + 1.875 y = 0.515388203 + 1.875 Combine like terms: 0.515388203 + 1.875 = 2.390388203 y = 2.390388203 Simplifying y = 2.390388203

Subproblem 2

y + -1.875 = -0.515388203 Simplifying y + -1.875 = -0.515388203 Reorder the terms: -1.875 + y = -0.515388203 Solving -1.875 + y = -0.515388203 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.875' to each side of the equation. -1.875 + 1.875 + y = -0.515388203 + 1.875 Combine like terms: -1.875 + 1.875 = 0.000 0.000 + y = -0.515388203 + 1.875 y = -0.515388203 + 1.875 Combine like terms: -0.515388203 + 1.875 = 1.359611797 y = 1.359611797 Simplifying y = 1.359611797

Solution

The solution to the problem is based on the solutions from the subproblems. y = {2.390388203, 1.359611797}

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