If α and β are the zeroes of the polynomial f(x)=ax²+bx+c = then the value of 1/α² + 1/β² will be:

Correct option is B

Given:

If α and β are the zeroes of the polynomial f(x) = ax²+bx+c 

Formula used:

Sum of α and β = $$\frac{-b}{a}$$

Product of α and β = $$\frac{c}{a}$$

Solution:

Sum of α and β = $$\frac{-b}{a}$$

Product of α and β = $$\frac{c}{a}$$

Now,

$$\frac{1}{\alpha^2} + \frac{1}{\beta^2} = \frac{\alpha^2 + \beta^2}{\alpha^2 \beta^2}$$

Add and substract (2αβ)

= $$\frac{\alpha^2 + \beta^2 + 2\alpha \beta - 2\alpha \beta}{\alpha^2 \beta^2}$$​​

$$= \frac{(\alpha + \beta)^2 - 2(\alpha \beta)}{(\alpha \beta)^2}$$

Put the values, then

​$$= \frac{\left(\frac{-b}{a}\right)^2 - 2 \left(\frac{c}{a}\right)}{\left(\frac{c}{a}\right)^2}$$

$$= \frac{\frac{b^2}{a^2} - 2\left(\frac{c}{a}\right)}{\frac{c^2}{a^2}}$$

= $$\left(\frac{b^2 - 2ac}{a^2}\right)\times{\frac{a^2}{c^2}}$$​​

$$= \frac{b^2 - 2ac}{c^2}$$

So, the value of 1/α²+ 1/β²will be $$\frac{b^2 - 2ac}{c^2}$$.

Thus, the correct answer is (b).

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