 # 3.9.2 Quadratic Functions, SPM Practice (Long Question)

Question 3:
Given that the quadratic function f(x) = 2x2px + p has a minimum value of –18 at x = 1.
(a) Find the values of p and q.
(b) With the value of p and q found in (a), find the values of x, where graph f(x) cuts the x-axis.
(c) Hence, sketch the graph of f(x).

Solution:
(a)
$\begin{array}{l}f\left(x\right)=2{x}^{2}-px+q\\ =2\left[{x}^{2}-\frac{p}{2}x+\frac{q}{2}\right]\\ =2\left[{\left(x+\frac{-p}{4}\right)}^{2}-{\left(\frac{-p}{4}\right)}^{2}+\frac{q}{2}\right]\\ =2\left[{\left(x-\frac{p}{4}\right)}^{2}-\frac{{p}^{2}}{16}+\frac{q}{2}\right]\\ =2{\left(x-\frac{p}{4}\right)}^{2}-\frac{{p}^{2}}{8}+q\end{array}$

(b)

(c) Question 4:
(a) Find the range of values of k if the equation x2kx + 3k – 5 = 0 does not have real roots.
(b) Show that the quadratic equation hx2 – (h + 3)x + 1 = 0 has real and distinc roots for all values of h.

Solution:
(a)

Graph function y = (k – 2)(k – 10) cuts the horizontal line at k = 2 and k = 10 when b2 – 4ac < 0. The range of values of k that satisfy the inequality above is 2 < k < 10.

(b)
$\begin{array}{l}h{x}^{2}-\left(h+3\right)x+1=0\\ {b}^{2}-4ac={\left(h+3\right)}^{2}-4\left(h\right)\left(1\right)\\ ={h}^{2}+6h+9-4h\\ ={h}^{2}+2h+9\\ ={\left(h+\frac{2}{2}\right)}^{2}-{\left(\frac{2}{2}\right)}^{2}+9\\ ={\left(h+1\right)}^{2}-1+9\\ ={\left(h+1\right)}^{2}+8\end{array}$

The minimum value of (h + 1) + 8 is 8, a positive value. Therefore, b2 – 4ac > 0 for all values of h.
Hence, quadratic equation hx2 – (h + 3)x + 1 = 0 has real and distinc roots for all values of h.