First slide
Theory of equations

If the roots of the equation x3+px2+qx1=0 form an increasing G.P., where p and q are real, then


Let the roots be a/r, a, ar, where a > 0, r > 1. Now,

a/r+a+ar=p                       (1)a(a/r)+a(ar)+(ar)(a/r)=q     (2)(a/r)(a)(ar)=1                           (3)

    a3=1    a=1

Hence, (3) is correct. From (1), putting a = 1, we get

p3>0             r+1r>2  p<3

Hence, (2) is not correct. Also,

1/r+1+r=p               (4)

From (2), putting a = 1, we get

1/r+r+1=q                   (5)

From (4) and (5), we have


Hence, (1) is correct. Now, as r > 1

a/r=1/r<1and   ar=r>1

Hence, (4) is correct.

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