Question

if alpha beta are the root of the qualification x square - 2 X - 1 = 0 find the value of Alpha square beta + beta square alpha​

Answer 1

Solution by the relation between roots and coefficients method :

The given quadratic equation is

   x² - 2x - 1 = 0 ...(i)

Since, α and β are the roots of (i) no. equation, by the relation between roots and coefficients we get

   α + β = - $\frac{-2}{1}$

   or, α + β = 2

   and αβ = - $\frac{1}{1}$

   or, αβ = - 1

Now, α²β + β²α

   = αβ (α + β)

   = - 1 * 2

   = - 2

Solution by finding the roots of the equation :

The given quadratic equation is

 x² - 2x - 1 = 0

 or, x² - 2x + 1 - 1 - 1 = 0

 or, (x - 1)² - 2 = 0

 or, (x - 1)² - (√2)² = 0

 or, (x - 1 + √2) (x - 1 - √2) = 0

Either x - 1 + √2 = 0 or, x - 1 - √2 = 0

   i.e., x = 1 - √2 , 1 + √2

Let, α = 1 - √2 and β = 1 + √2

Thus, a²β + β²α

     = αβ (α + β)

     = (1 - √2) (1 + √2) (1 - √2 + 1 + √2)

     = (1 - 2) * 2

     = - 2

Answer 2

Answer:

Step-by-step explanation:

Given that Alpha & Beta are the roots of this Quadratic Equation x² - 2x - 1 = 0.

Now,

Alpha + Beta = -b/a

=> Alpha + Beta = -(-2)/1

=> Alpha + Beta = 2.

& Alpha × Beta = c/a

=> Alpha × Beta = -1.

To Find:-

Alpha²Beta + Beta² Alpha

=> Alpha. beta ( Alpha + Beta)

=> -1 × 2

=> -2.