A Quadratic equations
B The discriminant of a quadratic
C Graphing quadratic functions
D Finding a quadratic from its graph
E Where functions meet
F Problem solving with quadratics
G Quadratic optimization
A quadratic equation is an equation of the form ax 2 +bx+c = 0 where a, b and c are constants, a ≠ 0.
A quadratic function is a function of the form f(x) = ax 2 +bx+c, a≠0.
Alternatively, it can be written as f : x | ax 2 +bx+c, a≠0.
Quadratic functions are members of the family of polynomials . f : x | ax 3 +bx 2 +cx+d, a≠0 is a cubic polynomial . f : x | ax 4 +bx 3 +cx 2 +dx+e, a≠0 is a quartic polynomial .
To solve quadratic equations we can:

Factorise the quadratic and use the Null Factor law : o If ab=0 then a=0 or b=0.

Complete the square

Use the quadratic formula

Use technology .
SOLVING BY FACTORISATION
Step 1: Make one side of the equation 0 by transferring all terms to one side. 3x 2 +5x=0.
Step 2: Fully factorise the other side. x(3x+5)
Step 3: Use the Null Factor law: If ab=0 then a=0 or b=0. x=0, or
3x+5=0.

Step 4: Solve the resulting linear equations. x=0, or x=-
5
3
.
SOLVING BY ‘COMPLETING THE SQUARE’
As you would be aware by now, not all quadratics factorise easily. For example, x 2 +4x+1 cannot be factorised by using a simple factorisation approach. This means that we need a different approach in order to solve x 2 +4x+1=0. One way is to use the ‘completing the square’ technique.
Equations of the form ax 2 +bx+c=0 can be converted to the form
(x+p) 2 =q from which the solutions are easy to obtain. x 2 +4x+1=0 x 2 +4x+4+1-4=0
(x+2) 2 -3=0
(x+2) 2 =3 𝑥 + 2 = √3 𝑥 = ±√3 − 2
THE QUADRATIC FORMULA
Many quadratic equations cannot be solved by factorising, and completing the square can be rather tedious. Consequently, the quadratic formula has been developed. This formula is: If ax 2 +bx+c=0, then 𝑥 =
−𝑏±√𝑏 2 −4𝑎𝑐
.
2𝑎 x 2 -2x-6=0 a=1, b=-2, c=-6 𝑥 =
−(−2) ± √(−2) 2
2(1)
− 4(1)(−6) 𝑥 = 𝑥 =
2 ± √4 + 24
2
2 ± √28
2 𝑥 =
2 ± 2√7
2 𝑥 = 1 ± √7

SOLVING USING TECHNOLOGY
You can use your graphics calculator to solve quadratic equations. If the right hand side is zero, you can graph the expression on the left hand side. The x-intercepts of the graph will be the solutions to the quadratic.
If the right hand side is non-zero, you can either:

Rearrange the equation so the right hand side is zero, then graph the expression and find the x-intercepts, or

Graph the expressions on the left and right hand sides on the same set of axes, then find the x-coordinates of the point where they meet.
In the quadratic formula, the quantity b 2 -4ac under the square root sign is called the discriminant .
The symbol delta is used to represent the discriminant, so delta= b 2 -4ac.
The quadratic formula becomes 𝑥 =
−𝑏±√𝑑𝑒𝑙𝑡𝑎
where delta replaces b
2𝑎
2 -
4ac.

If delta=0,
−𝑏
is the only solution (a repeated or double root )
2𝑎

If delta>0, √𝑑𝑒𝑙𝑡𝑎 is a positive real number, so there are two distinct real roots :
−𝑏+√𝑑𝑒𝑙𝑡𝑎
and
–𝑏−√𝑑𝑒𝑙𝑡𝑎
2𝑎 2𝑎

If delta < 0, √𝑑𝑒𝑙𝑡𝑎 is not a real number and so there are no real roots .
 If a, b and c are rational and ¢ is a perfect square then the equation has two rational roots which can be found by factorisation.
REVIEW OF TERMINOLOGY
The equation of a quadratic function is given by y = ax2 + bx + c, where a 6= 0. The graph of a quadratic function is called a parabola .
The point where the graph ‘turns’ is called the vertex .
If the graph opens upward, the y-coordinate of the vertex is the minimum and the graph is concave upwards. If the graph opens downward, the y-coordinate of the vertex is the maximum and the graph

is concave downwards. The vertical line that passes through the vertex is called the axis of symmetry . All parabolas are symmetrical about the axis of symmetry. The point where the graph crosses the y-axis is the y intercept . The points (if they exist) where the graph crosses the x-axis are called the x -intercepts , and correspond to the zeros of the function.
SKETCHING GRAPHS BY ‘COMPLETING THE SQUARE’
If we wish to find the vertex of a quadratic given in general form y = ax 2 +bx+c then one approach is to convert it to the form y=a(x-h) 2 +k where we can read off the vertex (h, k). To do this we may choose to
‘complete the square’.
Consider the simple case y = x 2 -4x+1, for which a=1. y = x 2 -4x+1 y = x 2 -4x+2 2 +1-2 2 y= (x-2) 2 -3
The graph of y = x 2 -4x+1 is the graph of y = x 2 translated through (
2
−3
).
THE DISCRIMINANT AND THE QUADRATIC GRAPH
Consider the graphs of: y = x 2 -2x+3, y=x 2 -2x+1, y=x 2 -2x-3.
All of these curves have the same axis of symmetry: x=1.
The discriminant delta determines if the graph:

Does not cut the x-axis (delta < 0)

Touches the x-axis (delta = 0)

Cuts the x-axis twice (delta > 0)
POSITIVE DEFINITE AND NEGATIVE DEFINITE QUADRATICS
Positive definite quadratics are quadratics which are positive for all values of x. So, ax 2 +bx+c>0 for all xisinthesetofallreals . Negative definite quadratics are quadratics which are negative for all values of x.
So, ax 2 +bx+c<0 for all xisinthesetofallreals . Tests:

A quadratic is positive definite if a > 0 and delta< 0.

A quadratic is negative definite if a < 0 and delta< 0.
If we are given sufficient information on or about a graph we can determine the quadratic function in whatever form is required.

Consider the graphs of a quadratic function and a linear function on the same set of axes.
Notice that we could have: 2, 1, or 0 points of intersections.
If the graphs meet, the coordinates of the points of intersection of the graphs of the two functions can be found by solving the two equations simultaneously.
When solving some problems algebraically, a quadratic equation results.
We are generally only interested in any real solutions which result. If the resulting quadratic equation has no real roots then the problem has no real solution. Any answer we obtain must be checked to see if it is reasonable. For example:

If we are finding a length then it must be positive and we reject any negative solutions

If we are finding ‘how many people are present’ then clearly the answer must be an integer.
We employ the following general problem solving method:
Step 1: If the information is given in words, translate it into algebra using a pronumeral such as x for the unknown. Write down the resulting equation.
Step 2: Solve the equation by a suitable method.
Step 3: Examine the solutions carefully to see if they are acceptable.
Step 4: Give your answer in a sentence.
For a quadratic function y = ax 2 +bx+c, we have already seen that:


If a > 0, the minimum value of y occurs at
If a < 0, the maximum value of y occurs at 𝑥 = 𝑥 =
−𝑏
2𝑎
−𝑏
.
2𝑎
The process of finding the maximum or minimum value of a function is called optimisation. Optimisation is a very useful tool when looking at such issues as maximising profits and minimising costs.

Golden Packet
Axis of symmetry of graph of a quadratic function 𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 − −> axis of symmetry 𝑥 =
−𝑏
2𝑎
Solution of a quadratic equation 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0 − −> 𝑥 =
−𝑏 ± √𝑏 2
2𝑎
Discriminant
Delta = b 2 -4ac
− 4𝑎𝑐
, 𝑎 ≠ 0