4x^2-x-21=0

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


Simplifying
4x2 + -1x + -21 = 0

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

Solving
-21 + -1x + 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 + -0.25x + x2 = 0

Move the constant term to the right:

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

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

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

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

The x term is -0.25x.  Take half its coefficient (-0.125).
Square it (0.015625) and add it to both sides.

Add '0.015625' to each side of the equation.
-0.25x + 0.015625 + x2 = 5.25 + 0.015625

Reorder the terms:
0.015625 + -0.25x + x2 = 5.25 + 0.015625

Combine like terms: 5.25 + 0.015625 = 5.265625
0.015625 + -0.25x + x2 = 5.265625

Factor a perfect square on the left side:
(x + -0.125)(x + -0.125) = 5.265625

Calculate the square root of the right side: 2.294694969

Break this problem into two subproblems by setting 
(x + -0.125) equal to 2.294694969 and -2.294694969.

Subproblem 1

x + -0.125 = 2.294694969 Simplifying x + -0.125 = 2.294694969 Reorder the terms: -0.125 + x = 2.294694969 Solving -0.125 + x = 2.294694969 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.125' to each side of the equation. -0.125 + 0.125 + x = 2.294694969 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + x = 2.294694969 + 0.125 x = 2.294694969 + 0.125 Combine like terms: 2.294694969 + 0.125 = 2.419694969 x = 2.419694969 Simplifying x = 2.419694969

Subproblem 2

x + -0.125 = -2.294694969 Simplifying x + -0.125 = -2.294694969 Reorder the terms: -0.125 + x = -2.294694969 Solving -0.125 + x = -2.294694969 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.125' to each side of the equation. -0.125 + 0.125 + x = -2.294694969 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + x = -2.294694969 + 0.125 x = -2.294694969 + 0.125 Combine like terms: -2.294694969 + 0.125 = -2.169694969 x = -2.169694969 Simplifying x = -2.169694969

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.419694969, -2.169694969}

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