4y^2-31y-24=0

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Solution for 4y^2-31y-24=0 equation:


Simplifying
4y2 + -31y + -24 = 0

Reorder the terms:
-24 + -31y + 4y2 = 0

Solving
-24 + -31y + 4y2 = 0

Solving for variable 'y'.

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

Divide each side by '4'.
-6 + -7.75y + y2 = 0

Move the constant term to the right:

Add '6' to each side of the equation.
-6 + -7.75y + 6 + y2 = 0 + 6

Reorder the terms:
-6 + 6 + -7.75y + y2 = 0 + 6

Combine like terms: -6 + 6 = 0
0 + -7.75y + y2 = 0 + 6
-7.75y + y2 = 0 + 6

Combine like terms: 0 + 6 = 6
-7.75y + y2 = 6

The y term is -7.75y.  Take half its coefficient (-3.875).
Square it (15.015625) and add it to both sides.

Add '15.015625' to each side of the equation.
-7.75y + 15.015625 + y2 = 6 + 15.015625

Reorder the terms:
15.015625 + -7.75y + y2 = 6 + 15.015625

Combine like terms: 6 + 15.015625 = 21.015625
15.015625 + -7.75y + y2 = 21.015625

Factor a perfect square on the left side:
(y + -3.875)(y + -3.875) = 21.015625

Calculate the square root of the right side: 4.584280205

Break this problem into two subproblems by setting 
(y + -3.875) equal to 4.584280205 and -4.584280205.

Subproblem 1

y + -3.875 = 4.584280205 Simplifying y + -3.875 = 4.584280205 Reorder the terms: -3.875 + y = 4.584280205 Solving -3.875 + y = 4.584280205 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '3.875' to each side of the equation. -3.875 + 3.875 + y = 4.584280205 + 3.875 Combine like terms: -3.875 + 3.875 = 0.000 0.000 + y = 4.584280205 + 3.875 y = 4.584280205 + 3.875 Combine like terms: 4.584280205 + 3.875 = 8.459280205 y = 8.459280205 Simplifying y = 8.459280205

Subproblem 2

y + -3.875 = -4.584280205 Simplifying y + -3.875 = -4.584280205 Reorder the terms: -3.875 + y = -4.584280205 Solving -3.875 + y = -4.584280205 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '3.875' to each side of the equation. -3.875 + 3.875 + y = -4.584280205 + 3.875 Combine like terms: -3.875 + 3.875 = 0.000 0.000 + y = -4.584280205 + 3.875 y = -4.584280205 + 3.875 Combine like terms: -4.584280205 + 3.875 = -0.709280205 y = -0.709280205 Simplifying y = -0.709280205

Solution

The solution to the problem is based on the solutions from the subproblems. y = {8.459280205, -0.709280205}

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