Question

What is the formula to find alpha minus beta?

Answer 1

We know, (a-b)² = (a+b)² - 4ab

So, (α - β)² = (α + β)² - 4αβ

α - β = √(α + β)² - 4αβ 

Answer 2

The formula for fine alpha minus beta is $\bold{\sqrt{\frac{\left(-b^{2}-4 a c\right)}{a}}}$

Solution:

Let us replace alpha as A and Beta as B

So we need to find what is alpha - beta i.e. A - B.

If we find the sum of roots, A + B we would get it as:

$A+B=\frac{-b}{a}$

$A B=\frac{c}{a}$

Now we know that,

$(A-B)^{2}=(A+B)^{2}-4 A B$

$=\left(\frac{-b}{a}\right)^{2}-4\left(\frac{c}{a}\right)$

$=\left(\frac{-b^{2}}{a^{2}}\right)-\frac{4 c}{a}$

$=\frac{\left(-b^{2}-4 a c\right)}{a^{2}}$

Now we need to take Square root on both sides. Thus, we get,

$\Rightarrow A-B=\sqrt{\frac{\left[-1\left(b^{2}+4 a c\right)\right]}{a}}$

Hence, the value of $\bold{A-B=\sqrt{\frac{\left(-b^{2}-4 a c\right)}{a}}}$