Now let's investigate the product of f(x) and g(x) when n = 0; therefore:

f(x) = -3

g(x) = 5

h(x) = -3 * 5 therefore h(x) = -15

 

Again we get a straight line...will this continue?

 

What happens when we make n=6

f(x) = 6x - 3

g(x) = 6x + 5

h(x) = (6x - 3) (6x + 5)

 

Here it looks like the product of two linear functions equals a quadratic function...will this be the trend?

 

Let's look once more with n=20

f(x) = 20x - 3

g(x) = 20x + 5

h(x) = (20x - 3) (20x + 5)

 

Yes, another quadratic function...let's break it down to see if this will always happen.

Suppose:

f(x) = ax + b

g(x) = cx + d

where a,b,c and d are real numbers

Letting h(x) = f(x) * g(x); h(x) = (ax + b) (cx + d)

Using the FOIL

h(x) = (ax)(cx) + (ax)d + (cx)b + bd

where h(x) = y

Since addition of real numbers yield a real number we can say

p = ac

q = (ad + bc)

r = bd

such that p,q, and r are real numbers

Therefore,

the general form of the quadratic equation

 


RETURN TO EXPLORING LINEAR EQUATIONS