Practice section 1 of 8

Express the following quadratic polynomials in the form $a(x+p)^2+q$, where $a,p,q$ are constants to be found.

(a) $x^2-5x-45$

(b) $3x^2+4x+6$

Find, in terms of $k$, the roots of the equation $x+\dfrac1x=k$.

Determine the nature of the roots of the equation $4x^2+6x-1=0$.

Find the range of values of $k$ for which $3x^2+6x+k$ is always positive for all real values of $x$.

Determine whether $x+1$ is a factor of the polynomial $x^3+x^2-2x-2$.

Simplify the following: (a) $\sqrt3\times\sqrt{27}$; (b) $\dfrac1{3-\sqrt2}$.