Question

If
$\alpha \: and \: \beta$
are the roots of equation,
$ax ^{2} + bx + c = 0$
, find the value of
$(1 + \frac{ \beta }{ \alpha } )(1 + \frac{ \alpha }{ \beta } )$
​

Answer 1

Given :-

α, β are the roots of the Quadratic equation ax² + bx + c =0

To find :-

value of (1 + β/α)(1 +α/β)

Explanation :-

As we know that,

In the Quadratic equation ax² + bx + c =0 sum of roots and product of roots is given by :—

α + β = -b/a --- eq❶ 

α β = c/a --- eq ❷

Now ,

(1 + β/α)(1 +α/β)

Simplifying the given

Since, taking the L.C.M to the denominators

1 + β/α = (α + β)/α

1 + α/β = ( α + β)/β

(1 + β/α)(1 +α/β)

→[ (α + β)/α ] [( α + β)/β ]

→ (α + β)² /α β

→ (-b/a)² / c/a [From eq -1 and 2]

→ (b²/a² )/ c/a

→ b² /a² × a/c

→ b²/ac

So, the value of (1 + β/α)(1 +α/β) = b²/ac