2x^2+19x+24=x

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


Simplifying
2x2 + 19x + 24 = x

Reorder the terms:
24 + 19x + 2x2 = x

Solving
24 + 19x + 2x2 = x

Solving for variable 'x'.

Reorder the terms:
24 + 19x + -1x + 2x2 = x + -1x

Combine like terms: 19x + -1x = 18x
24 + 18x + 2x2 = x + -1x

Combine like terms: x + -1x = 0
24 + 18x + 2x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(12 + 9x + x2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(12 + 9x + x2)' equal to zero and attempt to solve:

Simplifying
12 + 9x + x2 = 0

Solving
12 + 9x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-12' to each side of the equation.
12 + 9x + -12 + x2 = 0 + -12

Reorder the terms:
12 + -12 + 9x + x2 = 0 + -12

Combine like terms: 12 + -12 = 0
0 + 9x + x2 = 0 + -12
9x + x2 = 0 + -12

Combine like terms: 0 + -12 = -12
9x + x2 = -12

The x term is 9x.  Take half its coefficient (4.5).
Square it (20.25) and add it to both sides.

Add '20.25' to each side of the equation.
9x + 20.25 + x2 = -12 + 20.25

Reorder the terms:
20.25 + 9x + x2 = -12 + 20.25

Combine like terms: -12 + 20.25 = 8.25
20.25 + 9x + x2 = 8.25

Factor a perfect square on the left side:
(x + 4.5)(x + 4.5) = 8.25

Calculate the square root of the right side: 2.872281323

Break this problem into two subproblems by setting 
(x + 4.5) equal to 2.872281323 and -2.872281323.


Subproblem 1
x + 4.5 = 2.872281323

Simplifying
x + 4.5 = 2.872281323

Reorder the terms:
4.5 + x = 2.872281323

Solving
4.5 + x = 2.872281323

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-4.5' to each side of the equation.
4.5 + -4.5 + x = 2.872281323 + -4.5

Combine like terms: 4.5 + -4.5 = 0.0
0.0 + x = 2.872281323 + -4.5
x = 2.872281323 + -4.5

Combine like terms: 2.872281323 + -4.5 = -1.627718677
x = -1.627718677

Simplifying
x = -1.627718677


Subproblem 2
x + 4.5 = -2.872281323

Simplifying
x + 4.5 = -2.872281323

Reorder the terms:
4.5 + x = -2.872281323

Solving
4.5 + x = -2.872281323

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-4.5' to each side of the equation.
4.5 + -4.5 + x = -2.872281323 + -4.5

Combine like terms: 4.5 + -4.5 = 0.0
0.0 + x = -2.872281323 + -4.5
x = -2.872281323 + -4.5

Combine like terms: -2.872281323 + -4.5 = -7.372281323
x = -7.372281323

Simplifying
x = -7.372281323


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-1.627718677, -7.372281323}
Solution
x = {-1.627718677, -7.372281323}

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