Question

If 𝛼, 𝛽 are the zeroes of the polynomial f(x) = x2 + x – 2, then `(∝/β-∝/β)` 

 

Answer 1

By using the relationship between the zeroes of the quadratic polynomial. We have  

Sum of zeroes=`(-("Coefficient of x"))/("Coefficient of" x^2)` and Product of zeroes =`("Constant term")/("Coefficient of" x^2)` 

∴ 𝛼 + 𝛽 =`-1/1` and 𝛼𝛽 = `(-2)/1` 

⇒  𝛼 + 𝛽 = −1 and 𝛼𝛽 = −2 

Now, `(1/∝-1/β)^2=((β-∝)/(∝β ))` 

=`((∝+β)^2-4∝β)/(∝β)^2`       ` [∵(β-∝)^2=(∝+β)^2-4∝β]` 

=`((-1)^2-4(-2))/((-2)^2)`          `[∵∝+β=-1 and ∝β=-2]`

=`((-1)^2-4(-2))/4` 

=`9/4` 

∵` (1/∝-1/β)^2=9/4` 

⇒`1/∝-1/β=+-3/2`