Question

F(x) = -4x2 + 12x – 9

What is the value of the discriminant of f?
How many x-intercepts does the graph of f have?

Answer 1

`D = 0`

The graph has 1 x-intercept

Explanation:

Given

`f(x) = -4x^2 + 12x - 9`

Required

- Discriminant of f

- Number of x intercepts

Let D represent the discriminant;

D is calculated as thus

`D = b^2 - 4ac`

Where a, b and c are derived from the following general format;

`f(x) = ax^2 + bx +c`

By comparing
`f(x) = ax^2 + bx +c` with
`f(x) = -4x^2 + 12x - 9`

We have

`f(x) = f(x)\\ax^2 = -4x^2\\bx = 12x\\c = -9`

Solving further;

`a = -4\\b=12\\c=-9`

So, D can now be calculated;

`D = b^2 - 4ac` becomes

`D = 12^2 - 4 * -4 * -9`

`D = 144 - 144`

`D = 0`

Hence, the discriminant of f is 0

From the value of the discriminant, we can determine the number of x intercepts of the graph;

When D = 0, then; there exists only one x-intercept and it as calculated as thus

`x = (-b)/(2a)`

Recall that

`a = -4\\b=12\\c=-9`

So,
`x = (-b)/(2a)` becomes

`x = (-12)/(2 * -4)`

`x = (-12)/(-8)`

`x = (12)/(8)`

`x =1.5`