4x^2+x-21=0

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


Simplifying
4x2 + x + -21 = 0

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

Solving
-21 + x + 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 x.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

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

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

Combine like terms: 5.25 + 0.25 = 5.5
0.25 + 0.25x + x2 = 5.5

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 5.5

Calculate the square root of the right side: 2.34520788

Break this problem into two subproblems by setting 
(x + 0.5) equal to 2.34520788 and -2.34520788.

Subproblem 1

x + 0.5 = 2.34520788 Simplifying x + 0.5 = 2.34520788 Reorder the terms: 0.5 + x = 2.34520788 Solving 0.5 + x = 2.34520788 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = 2.34520788 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 2.34520788 + -0.5 x = 2.34520788 + -0.5 Combine like terms: 2.34520788 + -0.5 = 1.84520788 x = 1.84520788 Simplifying x = 1.84520788

Subproblem 2

x + 0.5 = -2.34520788 Simplifying x + 0.5 = -2.34520788 Reorder the terms: 0.5 + x = -2.34520788 Solving 0.5 + x = -2.34520788 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = -2.34520788 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -2.34520788 + -0.5 x = -2.34520788 + -0.5 Combine like terms: -2.34520788 + -0.5 = -2.84520788 x = -2.84520788 Simplifying x = -2.84520788

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.84520788, -2.84520788}

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