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algebra senior

Problem

If

r

and

s

are the roots of the equation

a x^{2} + b x + c = 0

, the value of

\frac{1}{r ^{2}} + \frac{1}{s ^{2}}

is:

(A)

b^{2} - 4 a c

(B)

\frac{b ^{2} - 4 a c}{2 a}

(C)

\frac{b ^{2} - 4 a c}{c ^{2}}

(D)

\frac{b ^{2} - 2 a c}{c ^{2}}

Solution

Note that

\frac{1}{r ^{2}} + \frac{1}{s ^{2}} = \frac{r ^{2} + s ^{2}}{r ^{2} s ^{2}} = \frac{( r + s ) ^{2} - 2 ( r s )}{( r s ) ^{2}}

. By Vieta's, this is

\frac{( - \frac{b}{a} ) ^{2} - 2 ( \frac{c}{a} )}{( \frac{c}{a} ) ^{2}} = \frac{\frac{b ^{2}}{a ^{2}} - \frac{2 c}{a}}{\frac{c ^{2}}{a ^{2}}} = \frac{b ^{2} - 2 a c}{c ^{2}} ⟹ \frac{b ^{2} - 2 a c}{c ^{2}}

Final answer

D