Answer:
[tex]D = 0[/tex]
The graph has 1 x-intercept
Step-by-step explanation:
Given
[tex]f(x) = -4x^2 + 12x - 9[/tex]
Required
- Discriminant of f
- Number of x intercepts
Let D represent the discriminant;
D is calculated as thus
[tex]D = b^2 - 4ac[/tex]
Where a, b and c are derived from the following general format;
[tex]f(x) = ax^2 + bx +c[/tex]
By comparing [tex]f(x) = ax^2 + bx +c[/tex] with [tex]f(x) = -4x^2 + 12x - 9[/tex]
We have
[tex]f(x) = f(x)\\ax^2 = -4x^2\\bx = 12x\\c = -9[/tex]
Solving further;
[tex]a = -4\\b=12\\c=-9[/tex]
So, D can now be calculated;
[tex]D = b^2 - 4ac[/tex] becomes
[tex]D = 12^2 - 4 * -4 * -9[/tex]
[tex]D = 144 - 144[/tex]
[tex]D = 0[/tex]
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
[tex]x = \frac{-b}{2a}[/tex]
Recall that
[tex]a = -4\\b=12\\c=-9[/tex]
So, [tex]x = \frac{-b}{2a}[/tex] becomes
[tex]x = \frac{-12}{2 * -4}[/tex]
[tex]x = \frac{-12}{-8}[/tex]
[tex]x = \frac{12}{8}[/tex]
[tex]x =1.5[/tex]
