Quadratic Equations

A polynomial function has a rule of the type $y = a_{n} x^{n} + a_{n - 1} + x^{n - 1} ... + a_{1} x^{1} + a_{0}$

where $n$ is a natural number or zero, and $a_{0}$ , $a_{1}$ , … , an are numbers called coefficients.

The degree of a polynomial is given by the value of n, the highest power of x with a non-zero coefficient

This page deals with polynomials of degree 2. These are called quadratic polynomials.

The graph of a linear polynomial function, y = mx + c, is a straight line and the graph of a quadratic polynomial function, $y = a x^{2} + b x + c, a \neq = 0$ is a parabola.

In order to sketch graphs of quadratics, we need to find the x-axis intercepts (if they exist), and to do this we need to solve quadratic equations. As an introduction to the methods of solving quadratic equations, the first two sections of this chapter review the basic algebraic processes of expansion and factorisation.

An algebraic expression is the sum of its terms. E.g., mx + c has terms mx and c

For expansions of the type $(a + b) (c + d)$ $a (c + d) + b (c + d) = a c + a d + b c + b d$

Perfect squares

$(a + b)^{2} = (a + b) (a + b) = a^{2} + 2 ab + b^{2}$

Differences of Squares

$(a - b) (a + b) = a^{2} - b^{2}$

Factorisation

Using Common Factors

If each term in an algebraic expression to be factorised contains a common factor, then this common factor is a factor of the entire expression.
To find the other factor, divide each term by the common factor. The common factor is placed outside the brackets.
This process is known as ‘taking the common factor outside the brackets’.

$k x^{2} + a x = x (k x + a)$

Grouping of terms

This method can be used for expressions containing four terms
Example: $x^{3} + 4 x^{2} - 3 x - 12 = (x^{3} + 4 x^{2}) - (3 x - 12) = x^{2} (x + 4) - 3 (x + 4) = (x^{2} - 3) (x + 4)$

Difference of two squares

You will recall the following identity from the previous section: $(a - b) (a + b) = a^{2} - b^{2}$
We can use this in the other way to factorise $a^{2} - b^{2} = (a - b) (a + b)$

Factorising Quadratic Polynomials

$x^{2} + 2 x y + y = (x + a) (x + b)$ , where $ab = y$ , and $a + b = 2 y$

Factorising the harder equations

$a x^{2} + b x + c$

Find $a c$
Find factors of $a c$ that add to $b$
Expand $b$ for those factors
Factorise and simplify

Solving Quadratic Equations

Write the equation in the form $a x^{2} + b x + c = 0$
Factorise the equation
Use the null factor theorem; i.e. if $mn = 0$ , it is implied that $m = 0$ and/or $n = 0$

Graphing Quadratic Equations

$y = a (x - b)^{2} + c$

b is how many units it is shifted right
a is the vertical dilation
c is how many units it is shifted up

Completing the square and turning points

To transpose a quadratic in polynomial form we can complete the square.
Consider a function $x^{2} + b x$
$x^{2} + b x + (\frac{b}{2})^{2} - (\frac{b}{2})^{2}$
$= (x + \frac{b}{2})^{2} - \frac{b ^{2}}{4}$