4x^2-2x-21=0

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


Simplifying
4x2 + -2x + -21 = 0

Reorder the terms:
-21 + -2x + 4x2 = 0

Solving
-21 + -2x + 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'.
-5.25 + -0.5x + x2 = 0

Move the constant term to the right:

Add '5.25' to each side of the equation.
-5.25 + -0.5x + 5.25 + x2 = 0 + 5.25

Reorder the terms:
-5.25 + 5.25 + -0.5x + x2 = 0 + 5.25

Combine like terms: -5.25 + 5.25 = 0.00
0.00 + -0.5x + x2 = 0 + 5.25
-0.5x + x2 = 0 + 5.25

Combine like terms: 0 + 5.25 = 5.25
-0.5x + x2 = 5.25

The x term is -0.5x.  Take half its coefficient (-0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
-0.5x + 0.0625 + x2 = 5.25 + 0.0625

Reorder the terms:
0.0625 + -0.5x + x2 = 5.25 + 0.0625

Combine like terms: 5.25 + 0.0625 = 5.3125
0.0625 + -0.5x + x2 = 5.3125

Factor a perfect square on the left side:
(x + -0.25)(x + -0.25) = 5.3125

Calculate the square root of the right side: 2.304886114

Break this problem into two subproblems by setting 
(x + -0.25) equal to 2.304886114 and -2.304886114.

Subproblem 1

x + -0.25 = 2.304886114 Simplifying x + -0.25 = 2.304886114 Reorder the terms: -0.25 + x = 2.304886114 Solving -0.25 + x = 2.304886114 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + x = 2.304886114 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + x = 2.304886114 + 0.25 x = 2.304886114 + 0.25 Combine like terms: 2.304886114 + 0.25 = 2.554886114 x = 2.554886114 Simplifying x = 2.554886114

Subproblem 2

x + -0.25 = -2.304886114 Simplifying x + -0.25 = -2.304886114 Reorder the terms: -0.25 + x = -2.304886114 Solving -0.25 + x = -2.304886114 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + x = -2.304886114 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + x = -2.304886114 + 0.25 x = -2.304886114 + 0.25 Combine like terms: -2.304886114 + 0.25 = -2.054886114 x = -2.054886114 Simplifying x = -2.054886114

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.554886114, -2.054886114}

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