Question

show that if the roots of the following quodratic
equation are
equal, then ad=bc x^2(a^2+b^2)+2(ac+bd)+(c^2+d^2)=0
​

Answer 1

Answer:

ad=bc

Step-by-step explanation:

for equal roots ,b²-4ac=0

a=(a²+b²) , b=2(ac+bd) , c=(c²+d²)

b²-4ac =0 = (2(ac+bd))²-4*(a²+b²)*(c²+d²)

= 4(a²c²+2abcd+b²d²)-4(a²c²+a²d²+b²c²+b²d²)=0

4(a²c²+2abcd+b²d²)=4(a²c²+a²d²+b²c²+b²d²)

 a²c²+2abcd+b²d²=a²c²+a²d²+b²c²+b²d²

           2abcd=a²d²+b²c²

      (ad)²+(bc)²-2abcd=0

by using identity,a²+b²-2ab=(a-b)²

we have ((ad)²-(bc))²=0

        (ad)²=(bc)²

taking square root

  ad=bc

hence proved