x^4-14x^3+53x^2=0

Solution for x^4-14x^3+53x^2=0 equation:

Simplifying
x4 + -14x3 + 53x2 = 0

Reorder the terms:
53x2 + -14x3 + x4 = 0

Solving
53x2 + -14x3 + x4 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), 'x2'.
x2(53 + -14x + x2) = 0

Subproblem 1
Set the factor 'x2' equal to zero and attempt to solve:

Simplifying
x2 = 0

Solving
x2 = 0

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

Simplifying
x2 = 0

Take the square root of each side:
x = {0}
Subproblem 2
Set the factor '(53 + -14x + x2)' equal to zero and attempt to solve:

Simplifying
53 + -14x + x2 = 0

Solving
53 + -14x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-53' to each side of the equation.
53 + -14x + -53 + x2 = 0 + -53

Reorder the terms:
53 + -53 + -14x + x2 = 0 + -53

Combine like terms: 53 + -53 = 0
0 + -14x + x2 = 0 + -53
-14x + x2 = 0 + -53

Combine like terms: 0 + -53 = -53
-14x + x2 = -53

The x term is -14x.  Take half its coefficient (-7).
Square it (49) and add it to both sides.

Add '49' to each side of the equation.
-14x + 49 + x2 = -53 + 49

Reorder the terms:
49 + -14x + x2 = -53 + 49

Combine like terms: -53 + 49 = -4
49 + -14x + x2 = -4

Factor a perfect square on the left side:
(x + -7)(x + -7) = -4

Can't calculate square root of the right side.

The solution to this equation could not be determined.

This subproblem is being ignored because a solution could not be determined.
Solution
x = {0}