 Hello friends, let's work out the following problem. It says if x-a is the GCD of x2-x-6 and x2-3x-18, find the value of a. So let's now move on to the solution and let px be the polynomial x2-x-6 and qx be the polynomial x2-3x-18. Now we are given that x-a is the GCD of px and qx, therefore x-a is a factor of px and qx, therefore by factor theorem p of a is 0 and q of a is 0. Now p of a is a square minus a minus 6 and this is equal to 0. Now let's factorize this quadratic equation for a. So we have a square minus 3a plus 2a minus 6 is equal to 0 taking a common from the first two terms. We have a into a minus 3 plus 2 into a minus 3 taking q common is equal to 0. So this implies a plus 2 into a minus 3 is equal to 0 taking a minus 3 common here. So this implies either a minus 3 is equal to 0 or a plus 2 is equal to 0 and this implies a is equal to 3 or a is equal to minus 2. Now we will use the fact that qa is 0. Now qa is a square plus 3a minus 18 and this is equal to 0. Now again we will factorize this quadratic equation. So we have a square plus 6a minus 3a minus 18 is equal to 0. So this implies taking a common from the first two terms. We have a plus 6 taking minus 3 common. We have minus 3 into a plus 6 is equal to 0. Now taking a plus 6 common we have a plus 6 into a minus 3 is equal to 0 and this implies a plus 6 is equal to 0 or a minus 3 is equal to 0. So this implies a is equal to minus 6 or a is equal to 3. Now we see that the only common value of a is coming out to be 3 and we know that x minus a is the greatest common divisor of the two polynomials px and qx therefore the only possible value of a is 3. So this completes the question and the session. Bye for now. Take care. Have a good day.