1. Show that, if the equations x^2 + 2px + q = 0 and x^2+2Px+Q=0 have a common root, then (q-Q)^2+4(P-p)(Pq-pQ)=0.
2. Find the conditions that the equations x^2+2x+a=0 and x^2+bx+3 should have a common root.
I have no clue how to do these problems.
Could someone please help???
Hmmm for 1. i can't seem to get the equation as shown in the equation. The equation is like (Q-q)^2/(2p-2P)^2 + 2P(Q - q)/(2p-2P) + Q = 0. When I expand or attempt to contract it, I dont seem to get anywhere closer to the answer.
For 2. How did u find out the values?
let a be common root therfor
we will apply the method of cross multiplication so
2p q 1 2p
2P Q 1 2P
(a^2)/(2pQ-2Pq) = a/(q-Q) = 1/(2P-2p)
a=(2pQ-2Pq) /(q-Q) = (q-Q)/(2P-2p)
now cross multiplying we get
NOTE : this is also the method to find the condition of common root so your second question will be solved in the same way.