5x^2-x=3

Solution for 5x^2-x=3 equation:

Simplifying
5x2 + -1x = 3

Reorder the terms:
-1x + 5x2 = 3

Solving
-1x + 5x2 = 3

Solving for variable 'x'.

Reorder the terms:
-3 + -1x + 5x2 = 3 + -3

Combine like terms: 3 + -3 = 0
-3 + -1x + 5x2 = 0

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

Divide each side by '5'.
-0.6 + -0.2x + x2 = 0

Move the constant term to the right:

Add '0.6' to each side of the equation.
-0.6 + -0.2x + 0.6 + x2 = 0 + 0.6

Reorder the terms:
-0.6 + 0.6 + -0.2x + x2 = 0 + 0.6

Combine like terms: -0.6 + 0.6 = 0.0
0.0 + -0.2x + x2 = 0 + 0.6
-0.2x + x2 = 0 + 0.6

Combine like terms: 0 + 0.6 = 0.6
-0.2x + x2 = 0.6

The x term is -0.2x.  Take half its coefficient (-0.1).
Square it (0.01) and add it to both sides.

Add '0.01' to each side of the equation.
-0.2x + 0.01 + x2 = 0.6 + 0.01

Reorder the terms:
0.01 + -0.2x + x2 = 0.6 + 0.01

Combine like terms: 0.6 + 0.01 = 0.61
0.01 + -0.2x + x2 = 0.61

Factor a perfect square on the left side:
(x + -0.1)(x + -0.1) = 0.61

Calculate the square root of the right side: 0.781024968

Break this problem into two subproblems by setting 
(x + -0.1) equal to 0.781024968 and -0.781024968.

Subproblem 1
x + -0.1 = 0.781024968

Simplifying
x + -0.1 = 0.781024968

Reorder the terms:
-0.1 + x = 0.781024968

Solving
-0.1 + x = 0.781024968

Solving for variable 'x'.

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

Add '0.1' to each side of the equation.
-0.1 + 0.1 + x = 0.781024968 + 0.1

Combine like terms: -0.1 + 0.1 = 0.0
0.0 + x = 0.781024968 + 0.1
x = 0.781024968 + 0.1

Combine like terms: 0.781024968 + 0.1 = 0.881024968
x = 0.881024968

Simplifying
x = 0.881024968

Subproblem 2
x + -0.1 = -0.781024968

Simplifying
x + -0.1 = -0.781024968

Reorder the terms:
-0.1 + x = -0.781024968

Solving
-0.1 + x = -0.781024968

Solving for variable 'x'.

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

Add '0.1' to each side of the equation.
-0.1 + 0.1 + x = -0.781024968 + 0.1

Combine like terms: -0.1 + 0.1 = 0.0
0.0 + x = -0.781024968 + 0.1
x = -0.781024968 + 0.1

Combine like terms: -0.781024968 + 0.1 = -0.681024968
x = -0.681024968

Simplifying
x = -0.681024968

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.881024968, -0.681024968}