x^2-24x-224=0

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Solution for x^2-24x-224=0 equation:


Simplifying
x2 + -24x + -224 = 0

Reorder the terms:
-224 + -24x + x2 = 0

Solving
-224 + -24x + x2 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '224' to each side of the equation.
-224 + -24x + 224 + x2 = 0 + 224

Reorder the terms:
-224 + 224 + -24x + x2 = 0 + 224

Combine like terms: -224 + 224 = 0
0 + -24x + x2 = 0 + 224
-24x + x2 = 0 + 224

Combine like terms: 0 + 224 = 224
-24x + x2 = 224

The x term is -24x.  Take half its coefficient (-12).
Square it (144) and add it to both sides.

Add '144' to each side of the equation.
-24x + 144 + x2 = 224 + 144

Reorder the terms:
144 + -24x + x2 = 224 + 144

Combine like terms: 224 + 144 = 368
144 + -24x + x2 = 368

Factor a perfect square on the left side:
(x + -12)(x + -12) = 368

Calculate the square root of the right side: 19.183326093

Break this problem into two subproblems by setting 
(x + -12) equal to 19.183326093 and -19.183326093.

Subproblem 1

x + -12 = 19.183326093 Simplifying x + -12 = 19.183326093 Reorder the terms: -12 + x = 19.183326093 Solving -12 + x = 19.183326093 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '12' to each side of the equation. -12 + 12 + x = 19.183326093 + 12 Combine like terms: -12 + 12 = 0 0 + x = 19.183326093 + 12 x = 19.183326093 + 12 Combine like terms: 19.183326093 + 12 = 31.183326093 x = 31.183326093 Simplifying x = 31.183326093

Subproblem 2

x + -12 = -19.183326093 Simplifying x + -12 = -19.183326093 Reorder the terms: -12 + x = -19.183326093 Solving -12 + x = -19.183326093 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '12' to each side of the equation. -12 + 12 + x = -19.183326093 + 12 Combine like terms: -12 + 12 = 0 0 + x = -19.183326093 + 12 x = -19.183326093 + 12 Combine like terms: -19.183326093 + 12 = -7.183326093 x = -7.183326093 Simplifying x = -7.183326093

Solution

The solution to the problem is based on the solutions from the subproblems. x = {31.183326093, -7.183326093}

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