4y^2-3y-9=0

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Solution for 4y^2-3y-9=0 equation:


Simplifying
4y2 + -3y + -9 = 0

Reorder the terms:
-9 + -3y + 4y2 = 0

Solving
-9 + -3y + 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'.
-2.25 + -0.75y + y2 = 0

Move the constant term to the right:

Add '2.25' to each side of the equation.
-2.25 + -0.75y + 2.25 + y2 = 0 + 2.25

Reorder the terms:
-2.25 + 2.25 + -0.75y + y2 = 0 + 2.25

Combine like terms: -2.25 + 2.25 = 0.00
0.00 + -0.75y + y2 = 0 + 2.25
-0.75y + y2 = 0 + 2.25

Combine like terms: 0 + 2.25 = 2.25
-0.75y + y2 = 2.25

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 = 2.25 + 0.140625

Reorder the terms:
0.140625 + -0.75y + y2 = 2.25 + 0.140625

Combine like terms: 2.25 + 0.140625 = 2.390625
0.140625 + -0.75y + y2 = 2.390625

Factor a perfect square on the left side:
(y + -0.375)(y + -0.375) = 2.390625

Calculate the square root of the right side: 1.54616461

Break this problem into two subproblems by setting 
(y + -0.375) equal to 1.54616461 and -1.54616461.

Subproblem 1

y + -0.375 = 1.54616461 Simplifying y + -0.375 = 1.54616461 Reorder the terms: -0.375 + y = 1.54616461 Solving -0.375 + y = 1.54616461 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 = 1.54616461 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + y = 1.54616461 + 0.375 y = 1.54616461 + 0.375 Combine like terms: 1.54616461 + 0.375 = 1.92116461 y = 1.92116461 Simplifying y = 1.92116461

Subproblem 2

y + -0.375 = -1.54616461 Simplifying y + -0.375 = -1.54616461 Reorder the terms: -0.375 + y = -1.54616461 Solving -0.375 + y = -1.54616461 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 = -1.54616461 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + y = -1.54616461 + 0.375 y = -1.54616461 + 0.375 Combine like terms: -1.54616461 + 0.375 = -1.17116461 y = -1.17116461 Simplifying y = -1.17116461

Solution

The solution to the problem is based on the solutions from the subproblems. y = {1.92116461, -1.17116461}

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