2.12.7 Quadratic Functions, SPM Practice (Paper 1)

Question 20:
Find the range of values of k if the quadratic equation 3(x² – kx – 1) = k – k² has two real and distinc roots.

Solution:
$\begin{array}{l}3\left(x^2-kx-1\right)=k-k^2\\ 3x^2-3kx-3-k+k^2=0\\ 3x^2-3kx+k^2-k-3=0\\ a=3,b=-3k,c=k^2-k-3\\ \\ \text{In cases of two real and distinc roots,}\\ b^2-4ac>0\text{ is applied}\text{.}\\ {\left(-3k\right)}^2-4\left(3\right)\left(k^2-k-3\right)>0\\ 9k^2-12k^2+12k+36>0\\ -3k^2+12k+36>0\\ -k^2+4k+12>0\\ k^2-4k-12<0\\ \left(k+2\right)\left(k-6\right)<0\\ k=-2,6\end{array}$



$\text{The range of values of }k\text{ is }-2<k<6.$

Question 21:
Given that the quadratic equation hx² – (h + 2)x – (h – 4) = 0 has real and distinc roots.  Find the range of values of h.

Solution:
$\begin{array}{l}\text{The quadratic equation }h{x}^2-\left(h+2\right)x-\left(h-4\right)=0\\ \text{has real and distinc roots}\text{.}\\ b^2-4ac>0\text{ is applied}\text{.}\\ {\left(-h-2\right)}^2-4\left(h\right)\left(-h+4\right)>0\\ h^2+4h+4+4h^2-16h>0\\ 5h^2-12h+4>0\\ \left(5h-2\right)\left(h-2\right)>0\\ \\ \text{The coefficient of }{h}^2\text{ is positive, }\\ \text{the region above the }x\text{-axis should be shaded}\text{.}\\ \left(5h-2\right)\left(h-2\right)=0\\ h=\frac{2}{5},2\end{array}$



$\begin{array}{l}\text{The range of values of }h\text{ for }\left(5h-2\right)\left(h-2\right)\text{>0 is}\\ h<\frac{2}{5}\text{ or }h>2.\end{array}$

Question 22:
The diagram below shows the graph of the quadratic function f(x) = (x + 3)² + 2h – 6, where h is a constant.


(a) State the equation of the axis of symmetry of the curve.
(b) Given the minimum value of the function is 4, find the value of h.

Solution:
(a)
When x + 3 = 0
 x = –3
Therefore, equation of the axis of symmetry of the curve is x = –3.

(b)
When x + 3 = 0, f(x) = 2h – 6
Minimum value of f(x) is 2h – 6.
2h – 6 = 4
2h = 10
h = 5