Practice section 6 of 8

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

(a) $2x^2-4x+5$

(b) $24-9x-x^2$

Find the exact roots of the equation $3x^2-6x+2=0$.

Show that $y=x^2-2x+4$ is positive for all real values of $x$.

Find the range of values of $k$ for which $4kx^2+4(2k+1)x+4k+1$ is always negative for all real values of $x$.

Solve the cubic equation $x^3-7x^2+4x+12=0$.

Simplify $\dfrac{23}{5-\sqrt2}-\sqrt2$.