Question

If $\alpha and \beta$ are the zeros of the quadratic polynomial $f(x)= x^{2}+x-2$ ,find the value of $\frac{1}{\alpha} - \frac{1}{\beta}$

Answer 1

SOLUTION:

Given : α and β are the zeroes of the quadratic polynomial f(x)= x² + x - 2

On comparing with ax² + bx + c,

a = 1 , b= 1 , c= -2

Sum of the zeroes = −coefficient of x / coefficient of x²

α + β  = -b/a = - 1/1 = -1

α+β = - 1 ……………………….(1)

Product of the zeroes = constant term/ Coefficient of x²

αβ = c/a = -2/1 = -2

αβ = - 2 ……………………….(2)

1/α  -  1/ β  

= (β–α)/αβ = -(α -  β) /αβ ……………….(3)

As we know that , (α - β)² = (α+β)² - 4αβ

(α - β)² = - 1² - 4 × -2 = 1 +8 = 9

(α - β)² = 9

(α - β) = √9 =  ±3

(α - β) = ±3 ………………(4)  

By Substituting the value from eq 2 & 4 ,in eq 3 , we get  

(β–α)/αβ = -(α -  β) /αβ

= - ( ±3 ) /-2 =  ±3/2  

1/α  -  1/ β   =  ±3/2  

Hence , the value of   1/α  -  1/ β  =  ± 3/2

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