4x^2-5x-22=0

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Solution for 4x^2-5x-22=0 equation:


Simplifying
4x2 + -5x + -22 = 0

Reorder the terms:
-22 + -5x + 4x2 = 0

Solving
-22 + -5x + 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'.
-5.5 + -1.25x + x2 = 0

Move the constant term to the right:

Add '5.5' to each side of the equation.
-5.5 + -1.25x + 5.5 + x2 = 0 + 5.5

Reorder the terms:
-5.5 + 5.5 + -1.25x + x2 = 0 + 5.5

Combine like terms: -5.5 + 5.5 = 0.0
0.0 + -1.25x + x2 = 0 + 5.5
-1.25x + x2 = 0 + 5.5

Combine like terms: 0 + 5.5 = 5.5
-1.25x + x2 = 5.5

The x term is -1.25x.  Take half its coefficient (-0.625).
Square it (0.390625) and add it to both sides.

Add '0.390625' to each side of the equation.
-1.25x + 0.390625 + x2 = 5.5 + 0.390625

Reorder the terms:
0.390625 + -1.25x + x2 = 5.5 + 0.390625

Combine like terms: 5.5 + 0.390625 = 5.890625
0.390625 + -1.25x + x2 = 5.890625

Factor a perfect square on the left side:
(x + -0.625)(x + -0.625) = 5.890625

Calculate the square root of the right side: 2.42706098

Break this problem into two subproblems by setting 
(x + -0.625) equal to 2.42706098 and -2.42706098.

Subproblem 1

x + -0.625 = 2.42706098 Simplifying x + -0.625 = 2.42706098 Reorder the terms: -0.625 + x = 2.42706098 Solving -0.625 + x = 2.42706098 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.625' to each side of the equation. -0.625 + 0.625 + x = 2.42706098 + 0.625 Combine like terms: -0.625 + 0.625 = 0.000 0.000 + x = 2.42706098 + 0.625 x = 2.42706098 + 0.625 Combine like terms: 2.42706098 + 0.625 = 3.05206098 x = 3.05206098 Simplifying x = 3.05206098

Subproblem 2

x + -0.625 = -2.42706098 Simplifying x + -0.625 = -2.42706098 Reorder the terms: -0.625 + x = -2.42706098 Solving -0.625 + x = -2.42706098 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.625' to each side of the equation. -0.625 + 0.625 + x = -2.42706098 + 0.625 Combine like terms: -0.625 + 0.625 = 0.000 0.000 + x = -2.42706098 + 0.625 x = -2.42706098 + 0.625 Combine like terms: -2.42706098 + 0.625 = -1.80206098 x = -1.80206098 Simplifying x = -1.80206098

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.05206098, -1.80206098}

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