Question

What is the formula for one upon alpha square plus one upon beta square

Answer 1

I derived the formula it is Beta sq -2ac divided by c Sq

Answer 2

Answer:

The formula for the expression is $\frac{1}{\alpha^2}+\frac{1}{\beta^2}=\frac{b^2-2ac}{c^2}$                  

Step-by-step explanation:

Given : Expression one upon alpha square plus one upon beta square.

To find : What is the formula of the expression ?

Solution :

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

Solving,

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

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

We know,

The zeros of the quadratic equation are

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

$\alpha \times \beta =\frac{c}{a}$ .....(2)

Squaring both side in (1),

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

$\alpha^2 + \beta^2+2\times \alpha\beta =\frac{b^2}{a^2}$

$\alpha^2 + \beta^2+2\times \frac{c}{a} =\frac{b^2}{a^2}$

$\alpha^2 + \beta^2=\frac{b^2}{a^2}-\frac{2c}{a}$

$\alpha^2 + \beta^2=\frac{b^2-2ac}{a^2}$......(3)

Substitute (2) and (3) in the expression,

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

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

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

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

Therefore, The formula for the expression is $\frac{1}{\alpha^2}+\frac{1}{\beta^2}=\frac{b^2-2ac}{c^2}$