4x^2-12x-52=0

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Solution for 4x^2-12x-52=0 equation: 


Simplifying
4x2 + -12x + -52 = 0

Reorder the terms:
-52 + -12x + 4x2 = 0

Solving
-52 + -12x + 4x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '4'.
4(-13 + -3x + x2) = 0

Ignore the factor 4.

Subproblem 1
Set the factor '(-13 + -3x + x2)' equal to zero and attempt to solve:

Simplifying
-13 + -3x + x2 = 0

Solving
-13 + -3x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '13' to each side of the equation.
-13 + -3x + 13 + x2 = 0 + 13

Reorder the terms:
-13 + 13 + -3x + x2 = 0 + 13

Combine like terms: -13 + 13 = 0
0 + -3x + x2 = 0 + 13
-3x + x2 = 0 + 13

Combine like terms: 0 + 13 = 13
-3x + x2 = 13

The x term is -3x.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3x + 2.25 + x2 = 13 + 2.25

Reorder the terms:
2.25 + -3x + x2 = 13 + 2.25

Combine like terms: 13 + 2.25 = 15.25
2.25 + -3x + x2 = 15.25

Factor a perfect square on the left side:
(x + -1.5)(x + -1.5) = 15.25

Calculate the square root of the right side: 3.905124838

Break this problem into two subproblems by setting 
(x + -1.5) equal to 3.905124838 and -3.905124838.


Subproblem 1
x + -1.5 = 3.905124838

Simplifying
x + -1.5 = 3.905124838

Reorder the terms:
-1.5 + x = 3.905124838

Solving
-1.5 + x = 3.905124838

Solving for variable 'x'.

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

Add '1.5' to each side of the equation.
-1.5 + 1.5 + x = 3.905124838 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + x = 3.905124838 + 1.5
x = 3.905124838 + 1.5

Combine like terms: 3.905124838 + 1.5 = 5.405124838
x = 5.405124838

Simplifying
x = 5.405124838


Subproblem 2
x + -1.5 = -3.905124838

Simplifying
x + -1.5 = -3.905124838

Reorder the terms:
-1.5 + x = -3.905124838

Solving
-1.5 + x = -3.905124838

Solving for variable 'x'.

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

Add '1.5' to each side of the equation.
-1.5 + 1.5 + x = -3.905124838 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + x = -3.905124838 + 1.5
x = -3.905124838 + 1.5

Combine like terms: -3.905124838 + 1.5 = -2.405124838
x = -2.405124838

Simplifying
x = -2.405124838


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {5.405124838, -2.405124838}
Solution
x = {5.405124838, -2.405124838}

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