The roots α and β of a quadratic equation are the square of two consecutive natural numbers. The geometric means of the two roots is 1 greater than the positive difference of the roots. If exactly one root of x2-k x+32=0 lies between α and β then find the number of integral value (s) of k.

Question

The roots α and β of a quadratic equation are the square of two consecutive natural numbers. The geometric means of the two roots is 1 greater than the positive difference of the roots. If exactly one root of x2-k x+32=0 lies between α and β then find the number of integral value (s) of k. (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

Let α= n 2, β=( n +1)2 √α β=|α-β|+1 Hence, n=2 ∴ α=4, β=9 Now, f(4) f(9) < 0 and also checking boundary points We get k ∈[12, (113/9))

Q. The roots α and β of a quadratic equation are the square of two consecutive natural numbers. The geometric means of the two roots is 1 greater than the positive difference of the roots. If exactly one root of x2−kx+32=0 lies between α and β then find the number of integral value (s) of k.

Let α=n2,β=(n+1)2 αβ​=∣α−β∣+1 Hence, n=2 ∴α=4,β=9 Now, f(4)f(9)<0 and also checking boundary points We get k∈[12,9113​)

Let $α = n^{2}, β = (n + 1)^{2}$
$α β = ∣ α - β ∣ + 1$
Hence, $n = 2$
$∴ α = 4, β = 9$
Now, $f (4) f (9) < 0$ and also checking boundary points
We get $k \in [12, \frac{113}{9})$