2x^2+41x-224=0

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Solution for 2x^2+41x-224=0 equation:


Simplifying
2x2 + 41x + -224 = 0

Reorder the terms:
-224 + 41x + 2x2 = 0

Solving
-224 + 41x + 2x2 = 0

Solving for variable 'x'.

Begin completing the square.  Divide all terms by
2 the coefficient of the squared term: 

Divide each side by '2'.
-112 + 20.5x + x2 = 0

Move the constant term to the right:

Add '112' to each side of the equation.
-112 + 20.5x + 112 + x2 = 0 + 112

Reorder the terms:
-112 + 112 + 20.5x + x2 = 0 + 112

Combine like terms: -112 + 112 = 0
0 + 20.5x + x2 = 0 + 112
20.5x + x2 = 0 + 112

Combine like terms: 0 + 112 = 112
20.5x + x2 = 112

The x term is 20.5x.  Take half its coefficient (10.25).
Square it (105.0625) and add it to both sides.

Add '105.0625' to each side of the equation.
20.5x + 105.0625 + x2 = 112 + 105.0625

Reorder the terms:
105.0625 + 20.5x + x2 = 112 + 105.0625

Combine like terms: 112 + 105.0625 = 217.0625
105.0625 + 20.5x + x2 = 217.0625

Factor a perfect square on the left side:
(x + 10.25)(x + 10.25) = 217.0625

Calculate the square root of the right side: 14.733041098

Break this problem into two subproblems by setting 
(x + 10.25) equal to 14.733041098 and -14.733041098.

Subproblem 1

x + 10.25 = 14.733041098 Simplifying x + 10.25 = 14.733041098 Reorder the terms: 10.25 + x = 14.733041098 Solving 10.25 + x = 14.733041098 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-10.25' to each side of the equation. 10.25 + -10.25 + x = 14.733041098 + -10.25 Combine like terms: 10.25 + -10.25 = 0.00 0.00 + x = 14.733041098 + -10.25 x = 14.733041098 + -10.25 Combine like terms: 14.733041098 + -10.25 = 4.483041098 x = 4.483041098 Simplifying x = 4.483041098

Subproblem 2

x + 10.25 = -14.733041098 Simplifying x + 10.25 = -14.733041098 Reorder the terms: 10.25 + x = -14.733041098 Solving 10.25 + x = -14.733041098 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-10.25' to each side of the equation. 10.25 + -10.25 + x = -14.733041098 + -10.25 Combine like terms: 10.25 + -10.25 = 0.00 0.00 + x = -14.733041098 + -10.25 x = -14.733041098 + -10.25 Combine like terms: -14.733041098 + -10.25 = -24.983041098 x = -24.983041098 Simplifying x = -24.983041098

Solution

The solution to the problem is based on the solutions from the subproblems. x = {4.483041098, -24.983041098}

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