1. Tardigrade
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  4. 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.

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Q. The roots $\alpha$ and $\beta$ 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 $x^{2}-k x+32=0$ lies between $\alpha$ and $\beta$ then find the number of integral value (s) of $k$.

Complex Numbers and Quadratic Equations

Solution:

Let $\alpha= n ^{2}, \beta=( n +1)^{2}$
$\sqrt{\alpha \beta}=|\alpha-\beta|+1$
Hence, $n=2$
$\therefore \alpha=4, \beta=9$
Now, $f(4) f(9) < 0$ and also checking boundary points
We get $k \in\left[12, \frac{113}{9}\right)$