53y^2-4y-48=0

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Solution for 53y^2-4y-48=0 equation:


Simplifying
53y2 + -4y + -48 = 0

Reorder the terms:
-48 + -4y + 53y2 = 0

Solving
-48 + -4y + 53y2 = 0

Solving for variable 'y'.

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

Divide each side by '53'.
-0.9056603774 + -0.07547169811y + y2 = 0

Move the constant term to the right:

Add '0.9056603774' to each side of the equation.
-0.9056603774 + -0.07547169811y + 0.9056603774 + y2 = 0 + 0.9056603774

Reorder the terms:
-0.9056603774 + 0.9056603774 + -0.07547169811y + y2 = 0 + 0.9056603774

Combine like terms: -0.9056603774 + 0.9056603774 = 0.0000000000
0.0000000000 + -0.07547169811y + y2 = 0 + 0.9056603774
-0.07547169811y + y2 = 0 + 0.9056603774

Combine like terms: 0 + 0.9056603774 = 0.9056603774
-0.07547169811y + y2 = 0.9056603774

The y term is -0.07547169811y.  Take half its coefficient (-0.03773584906).
Square it (0.001423994304) and add it to both sides.

Add '0.001423994304' to each side of the equation.
-0.07547169811y + 0.001423994304 + y2 = 0.9056603774 + 0.001423994304

Reorder the terms:
0.001423994304 + -0.07547169811y + y2 = 0.9056603774 + 0.001423994304

Combine like terms: 0.9056603774 + 0.001423994304 = 0.907084371704
0.001423994304 + -0.07547169811y + y2 = 0.907084371704

Factor a perfect square on the left side:
(y + -0.03773584906)(y + -0.03773584906) = 0.907084371704

Calculate the square root of the right side: 0.952409771

Break this problem into two subproblems by setting 
(y + -0.03773584906) equal to 0.952409771 and -0.952409771.

Subproblem 1

y + -0.03773584906 = 0.952409771 Simplifying y + -0.03773584906 = 0.952409771 Reorder the terms: -0.03773584906 + y = 0.952409771 Solving -0.03773584906 + y = 0.952409771 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.03773584906' to each side of the equation. -0.03773584906 + 0.03773584906 + y = 0.952409771 + 0.03773584906 Combine like terms: -0.03773584906 + 0.03773584906 = 0.00000000000 0.00000000000 + y = 0.952409771 + 0.03773584906 y = 0.952409771 + 0.03773584906 Combine like terms: 0.952409771 + 0.03773584906 = 0.99014562006 y = 0.99014562006 Simplifying y = 0.99014562006

Subproblem 2

y + -0.03773584906 = -0.952409771 Simplifying y + -0.03773584906 = -0.952409771 Reorder the terms: -0.03773584906 + y = -0.952409771 Solving -0.03773584906 + y = -0.952409771 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.03773584906' to each side of the equation. -0.03773584906 + 0.03773584906 + y = -0.952409771 + 0.03773584906 Combine like terms: -0.03773584906 + 0.03773584906 = 0.00000000000 0.00000000000 + y = -0.952409771 + 0.03773584906 y = -0.952409771 + 0.03773584906 Combine like terms: -0.952409771 + 0.03773584906 = -0.91467392194 y = -0.91467392194 Simplifying y = -0.91467392194

Solution

The solution to the problem is based on the solutions from the subproblems. y = {0.99014562006, -0.91467392194}

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