4x^2-21x-5=0

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Solution for 4x^2-21x-5=0 equation: 


Simplifying
4x2 + -21x + -5 = 0

Reorder the terms:
-5 + -21x + 4x2 = 0

Solving
-5 + -21x + 4x2 = 0

Solving for variable 'x'.

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

Divide each side by '4'.
-1.25 + -5.25x + x2 = 0

Move the constant term to the right:

Add '1.25' to each side of the equation.
-1.25 + -5.25x + 1.25 + x2 = 0 + 1.25

Reorder the terms:
-1.25 + 1.25 + -5.25x + x2 = 0 + 1.25

Combine like terms: -1.25 + 1.25 = 0.00
0.00 + -5.25x + x2 = 0 + 1.25
-5.25x + x2 = 0 + 1.25

Combine like terms: 0 + 1.25 = 1.25
-5.25x + x2 = 1.25

The x term is -5.25x.  Take half its coefficient (-2.625).
Square it (6.890625) and add it to both sides.

Add '6.890625' to each side of the equation.
-5.25x + 6.890625 + x2 = 1.25 + 6.890625

Reorder the terms:
6.890625 + -5.25x + x2 = 1.25 + 6.890625

Combine like terms: 1.25 + 6.890625 = 8.140625
6.890625 + -5.25x + x2 = 8.140625

Factor a perfect square on the left side:
(x + -2.625)(x + -2.625) = 8.140625

Calculate the square root of the right side: 2.853178053

Break this problem into two subproblems by setting 
(x + -2.625) equal to 2.853178053 and -2.853178053.


Subproblem 1
x + -2.625 = 2.853178053

Simplifying
x + -2.625 = 2.853178053

Reorder the terms:
-2.625 + x = 2.853178053

Solving
-2.625 + x = 2.853178053

Solving for variable 'x'.

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

Add '2.625' to each side of the equation.
-2.625 + 2.625 + x = 2.853178053 + 2.625

Combine like terms: -2.625 + 2.625 = 0.000
0.000 + x = 2.853178053 + 2.625
x = 2.853178053 + 2.625

Combine like terms: 2.853178053 + 2.625 = 5.478178053
x = 5.478178053

Simplifying
x = 5.478178053


Subproblem 2
x + -2.625 = -2.853178053

Simplifying
x + -2.625 = -2.853178053

Reorder the terms:
-2.625 + x = -2.853178053

Solving
-2.625 + x = -2.853178053

Solving for variable 'x'.

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

Add '2.625' to each side of the equation.
-2.625 + 2.625 + x = -2.853178053 + 2.625

Combine like terms: -2.625 + 2.625 = 0.000
0.000 + x = -2.853178053 + 2.625
x = -2.853178053 + 2.625

Combine like terms: -2.853178053 + 2.625 = -0.228178053
x = -0.228178053

Simplifying
x = -0.228178053


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {5.478178053, -0.228178053}

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