Question

Select the correct solution in each column of the table.

Solve the following equation.

Table included in image below:

Answer 1

No of real solutions =1

No of extraneous solution =2

Real solution: x =3

Explanation:

`(3)/(x)-(x)/(x+6)=(18)/(x^2+6x)`

solving:

Taking LCM of x, x+6 and x^2+6 we get x(x+6)

Multiply the equation with LCM

`(3)/(x)*x(x+6)-(x)/(x+6)*x(x+6)=(18)/(x^2+6x)*x(x+6)\\3(x+6)-x*x=(18)/(x(x+6))*x(x+6)\\3(x+6)-x*x=18\\3x+18-x^2=18\\-x^2+3x+18-18=0\\-x^2+3x=0\\x^2-3x=0\\x(x-3)=0\\x=0 \,\,and\,\, x =3\\`

Checking for extraneous solution

for extraneous solution we check the points where the solution is undefined

The solution will be undefined. if, x=0 or x=-6 so both are extraneous solutions

Putting x =3

`(3)/(3)-(3)/(3+6)=(18)/((3)^2+6(0))`

`(3)/(3)-(3)/(3+6)=(18)/((3)^2+6(3))\\1-(3)/(9)=(18)/(9+18)\\1-(1)/(3)=(18)/(27)\\(3-1)/(3)=(2)/(3)\\(2)/(3)=(2)/(3)`

So, x=3 is real solution.

Now, Selecting answers from tables

No of real solutions =1

No of extraneous solution =2

Real solution: x =3

Answer 2

Number of Number of Real solutions

real solution Extraneous solution

1 1 x=3

Explanation:

Extraneous solution--

It is a solution which is obtained on solving the equation but it does not satisfies the equation i.e. after it is put back to the equation it does not occur as a valid solution.

True solution or real solution--

It is the solution which is obtained on solving the equation and is also a valid solution to the equation.

The equation is:

`(3)/(x)-(x)/(x+6)=(18)/(x^2+6x)`

On taking lcm in the left hand side of the equation we get:

`(3* (x+6)-x* x)/(x(x+6))=(18)/(x^2+6x)\\\\i.e.\\\\(3x+18-x^2)/(x(x+6))=(18)/(x(x+6))\\\\i.e.\\\\(3x+18-x^2)/(x(x+6))-(18)/(x(x+6))=0\\\\i.e.\\\\(3x+18-x^2-18)/(x(x+6))=0\\\\i.e.\\\\(3x-x^2)/(x(x+6))=0\\\\i.e.\\\\3x-x^2=0\\\\i.e.\\\\x(3-x)=0\\\\i.e.\\\\x=0\ or\ x=3`

When we put x=0 back to the equation we observe that the first term of the left hand side of the equation becomes undefined.

Hence, x=0 is the extraneous solution.

whereas x=3 is a valid solution to the equation.