x^2+24x-19=0

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Solution for x^2+24x-19=0 equation:


Simplifying
x2 + 24x + -19 = 0

Reorder the terms:
-19 + 24x + x2 = 0

Solving
-19 + 24x + x2 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '19' to each side of the equation.
-19 + 24x + 19 + x2 = 0 + 19

Reorder the terms:
-19 + 19 + 24x + x2 = 0 + 19

Combine like terms: -19 + 19 = 0
0 + 24x + x2 = 0 + 19
24x + x2 = 0 + 19

Combine like terms: 0 + 19 = 19
24x + x2 = 19

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 = 19 + 144

Reorder the terms:
144 + 24x + x2 = 19 + 144

Combine like terms: 19 + 144 = 163
144 + 24x + x2 = 163

Factor a perfect square on the left side:
(x + 12)(x + 12) = 163

Calculate the square root of the right side: 12.767145335

Break this problem into two subproblems by setting 
(x + 12) equal to 12.767145335 and -12.767145335.

Subproblem 1

x + 12 = 12.767145335 Simplifying x + 12 = 12.767145335 Reorder the terms: 12 + x = 12.767145335 Solving 12 + x = 12.767145335 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 = 12.767145335 + -12 Combine like terms: 12 + -12 = 0 0 + x = 12.767145335 + -12 x = 12.767145335 + -12 Combine like terms: 12.767145335 + -12 = 0.767145335 x = 0.767145335 Simplifying x = 0.767145335

Subproblem 2

x + 12 = -12.767145335 Simplifying x + 12 = -12.767145335 Reorder the terms: 12 + x = -12.767145335 Solving 12 + x = -12.767145335 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 = -12.767145335 + -12 Combine like terms: 12 + -12 = 0 0 + x = -12.767145335 + -12 x = -12.767145335 + -12 Combine like terms: -12.767145335 + -12 = -24.767145335 x = -24.767145335 Simplifying x = -24.767145335

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.767145335, -24.767145335}

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