Question

if alpha and beta are zeros of x square - 5 x + 6 then find 3 upon 4 × *(Alpha square minus beta square)​

Answer 1

The value of   $\dfrac{3}{4}$ × ( α²  -  β² ) is $\dfrac{15}{4}$

Step-by-step explanation:

Given as :

The quadratic equation is   x² - 5 x + 6 = 0

The roots of this equation are  α  ,  β

Let The value of  $\dfrac{3}{4}$ × ( α²  -  β² ) =  k

According to question

For standard quadratic equation   ax² + b x + c = 0

The sum of roots = α + β = $\dfrac{-b}{a}$

The product of roots = α × β  = $\dfrac{c}{a}$

Now, compare standard equation with given quadratic equation

So,   α + β = $\dfrac{-(-5)}{1}$  = 5

The value of  α + β = 5

And  α × β  =  $\dfrac{6}{1}$  = 6

The value of   α × β = 6

Again

 ( α - β )²  =  ( α + β )² - 4 α × β

                =  (5 )² - 4 × 6

               = 25 - 24

               = 1

So, The value of α - β = 1

Now, A/Q

The value of   $\dfrac{3}{4}$ ( α²  -  β² ) =  k

Solving this equation

            k =  $\dfrac{3}{4}$ × [ ( α -  β )  ( α +  β ) ]

Or,        k = $\dfrac{3}{4}$ × [ ( 1 ) × ( 5 ) ]

Or,        k = $\dfrac{3}{4}$ × 5

∴           k = $\dfrac{15}{4}$

So, The value of   $\dfrac{3}{4}$ ( α²  -  β² ) =  k = $\dfrac{15}{4}$

Hence,  The value of   $\dfrac{3}{4}$ × ( α²  -  β² ) is $\dfrac{15}{4}$   Answer