Nature of the Roots (Combination of Straight Line and the Curve)
When you have a straight line and a curve, you can solve the equation of the straight line and the curve simultaneously and form a quadratic equation, ax² +bx + c = 0. The discriminant, $b^2-4ac$  gives information about the number of points of intersection.

Solving Quadratic Inequality

Solving Quadratic Inequality - Graph Method

Example 1
Express $y=5+4x-x^2$   in the form $y=a-{(x+b)}^2$   , where a, b, and c  are constants. Hence state the maximum  value of y and the value of x at which it occurs. Sketch the curve $y=5+4x-x^2$   .

Example
Sketch the curve of the quadratic function $f(x)=x^2-<!--\; -\; -->x-<!--\; -\; -->12$

Answer:
The shape of the graph
Since the coefficient of x² is positive, hence the graph is a U shape parabola with a minimum point.

The minimum point of the graph
By completing the square
$\begin{array}{l}f(x)=x^2-x-12\\ f(x)=x^2-x+{\left(\frac{1}{2}\right)}^2-{\left(\frac{1}{2}\right)}^2-12\\ f(x)={\left(x-\frac{1}{2}\right)}^2-\frac{1}{4}-12\\ f(x)={\left(x-\frac{1}{2}\right)}^2-12\frac{1}{4}\\ \\ \text{Minimum point = }(\frac{1}{2},-12\frac{1}{4})\end{array}$

For y-intercept, x = 0
$f(0)=(0{)}^2-<!--\; -\; -->(0)-<!--\; -\; -->12=-<!--\; -\; -->12$

For x-intercept, f(x) = 0
$\begin{array}{l}f(x)=x^2-x-12\\ 0=x^2-x-12\\ (x+5)(x-6)=0\\ x=-5\text{ or }x=6\end{array}$

Example 1 
Find the maximum or minimum value of each of the following quadratic function by completing the squares.  In each case, state the value of x at which the function is maximum or minimum.  And also, state the maximum or minimum point and axis of symmetry for each case.

(a) $f(x)=x^2+6x+7$
(b) $f(x)=2x^2-6x+7$
(c) $f(x)=5-2x-x^2$
(d) $f(x)=4+12x-3x^2$

Steps to convert general form of Quadratic Function into completing the square form

General form of quadratic function : $f(x)=a{x}^2+bx+c$

Completing the square form : $f(x)=a{(x+p)}^2+q$

Step 1 : Make sure the coefficient $x^2$ is 1, if not factorize.
Step 2 : Insert $+{\left(\frac{coefficient\text{}\text{}-of\text{}x}{2}\right)}^2-{\left(\frac{coefficient\text{}\text{}-of\text{}x}{2}\right)}^2$
Step 3 : Completing the square [convert $f(x)=a{x}^2+bx+c$ into $f(x)=a{(x+p)}^2+q$ .]

Example 1
State the maximum or minimum value of   for each of the following quadratic function and state the value of x at which the function is maximum or minimum.  Find the maximum or minimum point and finally state axis of symmetry for each case.

(a) $f(x)=-2{(x-3)}^2+4$
(b) $f(x)=3{(x-4)}^2+10$
(c) $f(x)=-3{(x+2)}^2-9$
(d) $f(x)=-8+2{(x+5)}^2$

Example:
Determine which of the following is a quadratic function.

1. $f(x)=(5x-3)(3x+8)$
2. $f(x)=2(3x+8)$
3. $f(x)=\frac{5}{2x^2}$

Answer:
(a)
  $\begin{array}{l}f(x)=(5x-3)(3x+8)\\ f(x)=15x^2+40x-9x-24\\ f(x)=15x^2+31x-24\end{array}$
Quadratic function

(b)
$\begin{array}{l}f(x)=2(3x+8)\\ f(x)=6x+16\end{array}$
Not quadratic function


(c) Not quadratic function