Question

if
$\alpha - \beta$
and beta are the zeros of a quadratic polynomial f x is equal to a x square + bx + c then evaluate Alpha minus beta

Answer 1

Final Answer :
Either α -β = √D/a or
-√D/a


Steps:
1) Given equation :
$f(x) = a {x}^{2} + bx + c \\ \alpha + \beta = \frac{ - b}{a} \\ \alpha \beta = \frac{c}{a}$

2)
${ (\alpha - \beta )}^{2} = {( \alpha + \beta )}^{2} - 4 \alpha \beta \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = > { (\frac{ - b}{a} )}^{2} - 4 \frac{c}{a} \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = > \frac{ {b}^{2} - 4ac }{ {a}^{2} } \:$

3) (α-β)^2= D/a^2
=>α-β = √D/a. or -√D/a


where D = Discriminant = b^2 -4ac

Answer 2

Answer:

Step-by-step explanation:

Final Answer :

Either α -β = √D/a or

-√D/a

Steps:

1) Given equation :

f(x) = a {x}^{2} + bx + c \\ \alpha + \beta = \frac{ - b}{a} \\ \alpha \beta = \frac{c}{a}  

2)

{ (\alpha - \beta )}^{2} = {( \alpha + \beta )}^{2} - 4 \alpha \beta \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = > { (\frac{ - b}{a} )}^{2} - 4 \frac{c}{a} \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = > \frac{ {b}^{2} - 4ac }{ {a}^{2} } \:  

3) (α-β)^2= D/a^2

=>α-β = √D/a. or -√D/a

where D = Discriminant = b^2 -4ac