Question

If $\alpha and \beta$ are the zeros of the quadratic polynomial $f(x)= x^{2} -5x+4$,find the value of $\frac{1}{\alpha} +\frac{1}{\beta} - 2 \alpha \beta$

Answer 1

SOLUTION :

Given : α and β are the roots of the quadratic polynomial. f(x)= x² - 5x + 4

On comparing with ax² + bx + c,

a = 1 , b= -5 , c= 4

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

α + β  = -b/a = -(-5)/1 = 5

α + β = 5 …………(1)

Product of the zeroes = constant term/ Coefficient of x²

αβ = c/a = 4/1 = 4

αβ = 4 ………..(2)

So,

1/α+1/β - 2αβ

=( β + α) / αβ - 2αβ

By Substituting the value from eq 1 & eq2 , we get  

= 5/4- 2×4

=5/ 4 - 8

= (5 - 32)/4

= −27/4

Hence, the value of  1/α+1/β - 2αβ is -27/4.

HOPE THIS ANSWER WILL HELP YOU….

Answer 2

Answer:

-27/4

Step-by-step explanation:

x² - 5x + 4 = 0

By Using Middle Term Splitting

x² - x - 4x + 4 = 0

x*( x - 1 ) - 4*( x - 1 )

(x - 4)*(x - 1)

So,

α = 4

β = 1

By Putting in 1/α + 1/β - 2αβ

1/4 + 1/1 - 2*4*1

5/4 - 8

= -27/4