If $x$ is a solution of the equation $\sqrt{2x + 1} - \sqrt{2x - 1} = 1$ for $x \ge \frac{1}{2}$,then $\sqrt{4x^2 - 1}$ is equal to:

Find a polynomial $f(x)$ of degree $2$ where $f(0)=8, f(1)=12, f(2)=18$.

Let $a, b$ be non-zero real numbers. Which of the following statements about the quadratic equation $a x^2+(a+b) x+b=0$ is necessarily true?
$I$. It has at least one negative root.
$II$. It has at least one positive root.
$III$. Both its roots are real.

If $f(x) = ax^{2} + bx + 2$ and $f(1) = 4, f(3) = 38$,then $a - b = $

If $x^2+x-6$ is a factor of $2x^3+x^2+ax+b$,then $6a+13b=$

The number of distinct real roots of the equation $x^{5}(x^{3}-x^{2}-x+1)+x(3x^{3}-4x^{2}-2x+4)-1=0$ is