5y^2+3y-75=0

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Solution for 5y^2+3y-75=0 equation:


Simplifying
5y2 + 3y + -75 = 0

Reorder the terms:
-75 + 3y + 5y2 = 0

Solving
-75 + 3y + 5y2 = 0

Solving for variable 'y'.

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

Divide each side by '5'.
-15 + 0.6y + y2 = 0

Move the constant term to the right:

Add '15' to each side of the equation.
-15 + 0.6y + 15 + y2 = 0 + 15

Reorder the terms:
-15 + 15 + 0.6y + y2 = 0 + 15

Combine like terms: -15 + 15 = 0
0 + 0.6y + y2 = 0 + 15
0.6y + y2 = 0 + 15

Combine like terms: 0 + 15 = 15
0.6y + y2 = 15

The y term is 0.6y.  Take half its coefficient (0.3).
Square it (0.09) and add it to both sides.

Add '0.09' to each side of the equation.
0.6y + 0.09 + y2 = 15 + 0.09

Reorder the terms:
0.09 + 0.6y + y2 = 15 + 0.09

Combine like terms: 15 + 0.09 = 15.09
0.09 + 0.6y + y2 = 15.09

Factor a perfect square on the left side:
(y + 0.3)(y + 0.3) = 15.09

Calculate the square root of the right side: 3.88458492

Break this problem into two subproblems by setting 
(y + 0.3) equal to 3.88458492 and -3.88458492.

Subproblem 1

y + 0.3 = 3.88458492 Simplifying y + 0.3 = 3.88458492 Reorder the terms: 0.3 + y = 3.88458492 Solving 0.3 + y = 3.88458492 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.3' to each side of the equation. 0.3 + -0.3 + y = 3.88458492 + -0.3 Combine like terms: 0.3 + -0.3 = 0.0 0.0 + y = 3.88458492 + -0.3 y = 3.88458492 + -0.3 Combine like terms: 3.88458492 + -0.3 = 3.58458492 y = 3.58458492 Simplifying y = 3.58458492

Subproblem 2

y + 0.3 = -3.88458492 Simplifying y + 0.3 = -3.88458492 Reorder the terms: 0.3 + y = -3.88458492 Solving 0.3 + y = -3.88458492 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.3' to each side of the equation. 0.3 + -0.3 + y = -3.88458492 + -0.3 Combine like terms: 0.3 + -0.3 = 0.0 0.0 + y = -3.88458492 + -0.3 y = -3.88458492 + -0.3 Combine like terms: -3.88458492 + -0.3 = -4.18458492 y = -4.18458492 Simplifying y = -4.18458492

Solution

The solution to the problem is based on the solutions from the subproblems. y = {3.58458492, -4.18458492}

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