4x^2-52x-40=0

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Solution for 4x^2-52x-40=0 equation: 


Simplifying
4x2 + -52x + -40 = 0

Reorder the terms:
-40 + -52x + 4x2 = 0

Solving
-40 + -52x + 4x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '4'.
4(-10 + -13x + x2) = 0

Ignore the factor 4.

Subproblem 1
Set the factor '(-10 + -13x + x2)' equal to zero and attempt to solve:

Simplifying
-10 + -13x + x2 = 0

Solving
-10 + -13x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '10' to each side of the equation.
-10 + -13x + 10 + x2 = 0 + 10

Reorder the terms:
-10 + 10 + -13x + x2 = 0 + 10

Combine like terms: -10 + 10 = 0
0 + -13x + x2 = 0 + 10
-13x + x2 = 0 + 10

Combine like terms: 0 + 10 = 10
-13x + x2 = 10

The x term is -13x.  Take half its coefficient (-6.5).
Square it (42.25) and add it to both sides.

Add '42.25' to each side of the equation.
-13x + 42.25 + x2 = 10 + 42.25

Reorder the terms:
42.25 + -13x + x2 = 10 + 42.25

Combine like terms: 10 + 42.25 = 52.25
42.25 + -13x + x2 = 52.25

Factor a perfect square on the left side:
(x + -6.5)(x + -6.5) = 52.25

Calculate the square root of the right side: 7.228416147

Break this problem into two subproblems by setting 
(x + -6.5) equal to 7.228416147 and -7.228416147.


Subproblem 1
x + -6.5 = 7.228416147

Simplifying
x + -6.5 = 7.228416147

Reorder the terms:
-6.5 + x = 7.228416147

Solving
-6.5 + x = 7.228416147

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '6.5' to each side of the equation.
-6.5 + 6.5 + x = 7.228416147 + 6.5

Combine like terms: -6.5 + 6.5 = 0.0
0.0 + x = 7.228416147 + 6.5
x = 7.228416147 + 6.5

Combine like terms: 7.228416147 + 6.5 = 13.728416147
x = 13.728416147

Simplifying
x = 13.728416147


Subproblem 2
x + -6.5 = -7.228416147

Simplifying
x + -6.5 = -7.228416147

Reorder the terms:
-6.5 + x = -7.228416147

Solving
-6.5 + x = -7.228416147

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '6.5' to each side of the equation.
-6.5 + 6.5 + x = -7.228416147 + 6.5

Combine like terms: -6.5 + 6.5 = 0.0
0.0 + x = -7.228416147 + 6.5
x = -7.228416147 + 6.5

Combine like terms: -7.228416147 + 6.5 = -0.728416147
x = -0.728416147

Simplifying
x = -0.728416147


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {13.728416147, -0.728416147}
Solution
x = {13.728416147, -0.728416147}

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