The equation has one real solution (x = 3) and one extraneous solution (x = 0), making the total number of real solutions equal to one.
To solve the given equation "3/x - x / x+6 = 18 / x^2 + 6x", let's find a common denominator, which is x(x+6). Multiplying both sides by this common denominator, we get:
3(x+6) - x^2 = 18
Simplifying further:
3x + 18 - x^2 = 18
Rearranging terms:
x^2 - 3x = 0
Factoring out x:
x(x - 3) = 0
This equation has two potential solutions: x = 0 and x = 3. Now, let's check for extraneous solutions, which may occur if a value makes the original equation undefined.
For x = 0, the original equation becomes undefined due to division by zero in the term 3/x. Therefore, x = 0 is an extraneous solution.
For x = 3, the original equation is defined, and substituting x = 3 back into the original equation validates x = 3 as a valid solution.
Now, looking at the table options:
Number of Real Solutions: 1 (since x = 3 is a valid solution).
Number of Extraneous Solutions: 1 (since x = 0 is an extraneous solution).
Real Solutions: x = 3.
Extraneous Solutions: x = 0.
The question probable may be:
Select the correct solution in each column of the table.
Solve the following equation
3/x - x / x+6 = 18 / x^2 +6x
Examine the given table , and select the accurate number of each type of solution as well as any real solutions that exist.
Number of Number of Real solutions
Real solution Extraneous
Solutions
0 0 x = 0
1 1 x = 3
2 2 x = 0 , x=3