Question

if alpha and beta are zeros of the polynomial x^2 -ax + b,then find the values of i) alpha^2 + beta ^2 ii) 1 /alpha + 1 /beta

Answer 1

Answer:

(i) $a^2-2b$

(ii) $\frac{a}{b}$

Step-by-step explanation:

Since, if p and q are roots of a quadratic equation,

ax² + bx + c

Then,

$p+q=-\frac{b}{a}$

$p.q=\frac{c}{a}$

Here, the given quadratic equation,

$x^2-ax+b$

If $\alpha$ and $\beta$ are roots of the given equation,

So, by the above property,

$\alpha+\beta=-\frac{-a}{1}=a$---------(1)

$\alpha.\beta=\frac{b}{1}=b$ --------(2),

(i) By squaring equation (1),

$(\alpha+\beta)^2=a^2$

$(\alpha)^2+(\beta)^2+2\alpha.\beta = a^2$

From equation (2),

$(\alpha)^2+(\beta)^2+2b = a^2$

$\implies (\alpha)^2+(\beta)^2= a^2-2b$

(ii) $\frac{1}{\alpha}+\frac{1}{\beta}=\frac{\beta+\alpha}{\alpha.\beta}=\frac{a}{b}$