x^2+3x+24=47

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Solution for x^2+3x+24=47 equation: 


Simplifying
x2 + 3x + 24 = 47

Reorder the terms:
24 + 3x + x2 = 47

Solving
24 + 3x + x2 = 47

Solving for variable 'x'.

Reorder the terms:
24 + -47 + 3x + x2 = 47 + -47

Combine like terms: 24 + -47 = -23
-23 + 3x + x2 = 47 + -47

Combine like terms: 47 + -47 = 0
-23 + 3x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '23' to each side of the equation.
-23 + 3x + 23 + x2 = 0 + 23

Reorder the terms:
-23 + 23 + 3x + x2 = 0 + 23

Combine like terms: -23 + 23 = 0
0 + 3x + x2 = 0 + 23
3x + x2 = 0 + 23

Combine like terms: 0 + 23 = 23
3x + x2 = 23

The x term is 3x.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3x + 2.25 + x2 = 23 + 2.25

Reorder the terms:
2.25 + 3x + x2 = 23 + 2.25

Combine like terms: 23 + 2.25 = 25.25
2.25 + 3x + x2 = 25.25

Factor a perfect square on the left side:
(x + 1.5)(x + 1.5) = 25.25

Calculate the square root of the right side: 5.024937811

Break this problem into two subproblems by setting 
(x + 1.5) equal to 5.024937811 and -5.024937811.


Subproblem 1
x + 1.5 = 5.024937811

Simplifying
x + 1.5 = 5.024937811

Reorder the terms:
1.5 + x = 5.024937811

Solving
1.5 + x = 5.024937811

Solving for variable 'x'.

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

Add '-1.5' to each side of the equation.
1.5 + -1.5 + x = 5.024937811 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + x = 5.024937811 + -1.5
x = 5.024937811 + -1.5

Combine like terms: 5.024937811 + -1.5 = 3.524937811
x = 3.524937811

Simplifying
x = 3.524937811


Subproblem 2
x + 1.5 = -5.024937811

Simplifying
x + 1.5 = -5.024937811

Reorder the terms:
1.5 + x = -5.024937811

Solving
1.5 + x = -5.024937811

Solving for variable 'x'.

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

Add '-1.5' to each side of the equation.
1.5 + -1.5 + x = -5.024937811 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + x = -5.024937811 + -1.5
x = -5.024937811 + -1.5

Combine like terms: -5.024937811 + -1.5 = -6.524937811
x = -6.524937811

Simplifying
x = -6.524937811


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {3.524937811, -6.524937811}

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