Given :- x² - x - 2 = 0
The given equation is of the form
ax² + bx + c = 0
Here , a = 1 , b = -1 , c = - 2

Correct Option: B

Given :- x² - x - 2 = 0
The given equation is of the form
ax² + bx + c = 0
Here , a = 1 , b = -1 , c = - 2
Now, D² = b² - 4ac = ( -1 )² - 4 x 1 x ( - 2 )
D = √9 = 3
So, roots are rational .
Hence, both the roots must be integers.

Given that :- √7x² - 6x - 13√7 = 0
⇒ √7x² - 13x + 7x - 13√7 = 0
⇒ x( √7x - 13 ) + √7( √7x - 13 ) = 0
⇒ ( x + √7 ) ( √7x - 13 )
⇒ x + √7 = 0 or √7x - 13 = 0

Correct Option: C

Given that :- √7x² - 6x - 13√7 = 0
⇒ √7x² - 13x + 7x - 13√7 = 0
⇒ x( √7x - 13 ) + √7( √7x - 13 ) = 0
⇒ ( x + √7 ) ( √7x - 13 )
⇒ x + √7 = 0 or √7x - 13 = 0

The given quadratic equation is
3x² - 5x + P = 0
Comparing with ax ² + bx + c = 0, we get
a = 0, b = - 5, c = P
If the given quadratic equation has equal roots then its discriminant = 0

Correct Option: C

The given quadratic equation is
3x ² - 5x + P = 0
Comparing with ax ² + bx + c = 0, we get
a = 0, b = - 5, c = P
If the given quadratic equation has equal roots then its discriminant = 0
⇒ b ² - 4ac = 0
⇒ ( - 5 ) ² - 4 x (3) x (P) = 0
⇒ 25 - 12P = 0

As per the given above question , we have
Let, x = 6 + √6 + √6 + √6 +........
On squaring both sides, we have
x² = 6 + √6 + √6 + √6 +........
⇒ x² = 6 + x
⇒ x² - x - 6 = 0

Correct Option: D

As per the given above question , we have
Let, x = 6 + √6 + √6 + √6 +........
On squaring both sides, we have
x² = 6 + √6 + √6 + √6 +........
⇒ x² = 6 + x
⇒ x² - x - 6 = 0
⇒ x² - 3x + 2x - 6 = 0
⇒ x( x - 3 ) + 2( x - 3 ) = 0
⇒ ( x - 3 ) ( x + 2 ) = 0
⇒ x = 3 and x ≠ −2 because numbers are positive.
Thus , correct answer will be 3 .

According to question ,
∵ α, β are the roots of the equation x² - 8x + 1 = 0

Correct Option: A

According to question ,
∵ α, β are the roots of the equation x² - 8x + 1 = 0