4x^2+21x-1=0

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Solution for 4x^2+21x-1=0 equation:


Simplifying
4x2 + 21x + -1 = 0

Reorder the terms:
-1 + 21x + 4x2 = 0

Solving
-1 + 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'.
-0.25 + 5.25x + x2 = 0

Move the constant term to the right:

Add '0.25' to each side of the equation.
-0.25 + 5.25x + 0.25 + x2 = 0 + 0.25

Reorder the terms:
-0.25 + 0.25 + 5.25x + x2 = 0 + 0.25

Combine like terms: -0.25 + 0.25 = 0.00
0.00 + 5.25x + x2 = 0 + 0.25
5.25x + x2 = 0 + 0.25

Combine like terms: 0 + 0.25 = 0.25
5.25x + x2 = 0.25

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 = 0.25 + 6.890625

Reorder the terms:
6.890625 + 5.25x + x2 = 0.25 + 6.890625

Combine like terms: 0.25 + 6.890625 = 7.140625
6.890625 + 5.25x + x2 = 7.140625

Factor a perfect square on the left side:
(x + 2.625)(x + 2.625) = 7.140625

Calculate the square root of the right side: 2.672194791

Break this problem into two subproblems by setting 
(x + 2.625) equal to 2.672194791 and -2.672194791.

Subproblem 1

x + 2.625 = 2.672194791 Simplifying x + 2.625 = 2.672194791 Reorder the terms: 2.625 + x = 2.672194791 Solving 2.625 + x = 2.672194791 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 = 2.672194791 + -2.625 Combine like terms: 2.625 + -2.625 = 0.000 0.000 + x = 2.672194791 + -2.625 x = 2.672194791 + -2.625 Combine like terms: 2.672194791 + -2.625 = 0.047194791 x = 0.047194791 Simplifying x = 0.047194791

Subproblem 2

x + 2.625 = -2.672194791 Simplifying x + 2.625 = -2.672194791 Reorder the terms: 2.625 + x = -2.672194791 Solving 2.625 + x = -2.672194791 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 = -2.672194791 + -2.625 Combine like terms: 2.625 + -2.625 = 0.000 0.000 + x = -2.672194791 + -2.625 x = -2.672194791 + -2.625 Combine like terms: -2.672194791 + -2.625 = -5.297194791 x = -5.297194791 Simplifying x = -5.297194791

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.047194791, -5.297194791}

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