Quadratic Equation with 'p' and 'q'

A quadratic equation in 'p' and 'q' is a factored quadratic equation which is written as follows:

`y = a(x - p)(x - q)`

In the above quadratic equation, the variables are y and x whereas the constants are a, p and q, that is, in place of a, p and q you will generally find numbers there.

For example, the equation

`y = (x + 2)(x - 3)`

is a quadratic equation in which a = 1, p = -2 and q = 3

Characteristics of a quadratic equation with p and q

A quadratic equation with p and q has the following characteristics:

• 'p' and 'q' are the x-intercepts, the roots or what you could also call the solutions of the quadratic equation
• 'a' is a constant value which tells you a number of things about the quadratic equation. These are discussed below

p and q are the x-intercepts or solutions

p and q are the x-intercepts or solutions of the quadratic equation. For example, consider the quadratic equation (x + 4)(x - 4) = 0. Its solutions are -4 and 4 because the numbers in place of p and q in this equation are -4 and 4.

Now consider the following quadratic function

`y = (x + 2)(x - 3)`

Its graph is shown below. It clearly has x-intercepts at x = -2 and x = 3.

p and q are the x-intercepts

It must however be kept in mind that the sign of p and q are opposite to the sign which appears in the equation. For example, when you compare the equation

`y = (x + 2)(x - 3)`

with the equation

`y = a(x - p)(x - q)` 

you get

`x + 2 = x - p`
`=> 2 = -p`
`=> p = -2`

and you also get

`x - 3 = x - q`
`=> -3 = -q`
`=> q = 3`

Clearly, the values of p and q have opposite signs to the numbers in their places in the equation. Remembering this helps you make less errors while solving such quadratic equation problems.

Importance of 'a' 

The 'a' in the quadratic equation `y = a(x - p)(x - q)` plays an important role as it defines more than one characteristics of the parabola, which are covered one by one below:

Facing up/down or right/left

A parabola faces upwards or right when the value of 'a' is positive. The following images show the graphs of two quadratic equations in which the value of `a` is positive (a is equal to 1 in both the graphs)

Graph of y = (x + 1)(x - 1) where a = 1
Parabola facing up

Graph of x = (y + 1)(y - 1) where a = 1
Parabola is facing right

On the other hand, when 'a' has a negative value, the parabola, that is, the graph of the quadratic equation, faces down or left as shown in the graphs below:

Graph of y = -(x + 1)(x - 1) where a = -1
Parabola is facing left

Graph of x = -(y + 1)(y - 1) where a = -1
Parabola is facing down

Calculating the length of latus rectum

The value of 'a' is useful in obtaining the length of the latus rectum and other things such as coordinates of focus, equation of directrix and equation of axis of the parabola (For more information on the latus rectum, focus, directrix and axis, please visit this post). Let us first consider the latus rectum. For any value of 'a' in a quadratic equation, the length of the latus rectum is equal to four times the value of 'a'. This is explained in detail here.

Calculating the distance of the focus from the vertex, or the distance of the vertex from the directrix

In general, the distance of the focus from the vertex of a parabola is equal to 'a'. Furthermore the perpendicular distance of the directrix from the vertex is also equal to 'a'.