Show that the equations x 2 + p x + q = 0 and x 2 + q x + p = 0 have a common root if p = q or p + q + 1 = 0

Question

Show that the equations       x    2    +  p  x  +  q  =  0 and       x    2    +  q  x  +  p  =  0 have a common root if   p  =  q or   p  +  q  +  1  =  0

trollbabyfeetwa · Accepted Answer

Step 1Let the 2 eqns have root &#039;a&#039; common to thema)       a    2    +  p  a  +  q  =  0b)       a    2    +  q  a  +  p  =  0solve  (  p  −  q  )  a  +  (  p  −  q  )  =  0    ⇒  p  =  qStep 2let eqn1 have roots a,b and eqn 2 have roots a,c1)   a  +  b  =  −  p2)   a  +  c  =  −  qfrom (1) and (2)3)   b  −  c  =  q  −  p4)   a  b  =  q5)   a  c  =  pfrom (4) and (5)6)       b    c    =      q    p  from (3) and (6)  b  =  q and   c  =  psubstituting b=qin first eqn:  p  +  q  +  1  =  0

Show that the equations x^2+px+q=0 and x^2+qx+p=0 have a common root if p=q or p+q+1=0

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