For this case we have the following function:
[tex]f (x) = - 4x ^ 2 + 12x-9[/tex]
[tex]y = 0[/tex] we have:
[tex]-4x ^ 2 + 12x-9 = 0[/tex]
Where:
[tex]a = -4\\b = 12\\c = -9[/tex]
By definition, the discriminant of a quadratic equation is given by:
[tex]d = b ^ 2-4 (a) (c)[/tex]
[tex]d> 0:[/tex] Two different real roots
[tex]d = 0:[/tex] Two equal real roots
[tex]d <0[/tex]: Two different complex roots
Substituting the values we have:
[tex]d = (12) ^ 2-4 (-4) (- 9)\\d = 144-144[/tex]
[tex]d = 0[/tex]
We have two equal real roots.
To find the intersections with the x axis, we do [tex]y = 0:[/tex]
[tex]-4x ^ 2 + 12x-9 = 0[/tex]
We apply the quadratic formula:
[tex]x = \frac {-b \pm \sqrt {b ^ 2-4 (a) (c)}} {2a}[/tex]
Substituting the values we have:
[tex]x = \frac {-12 \pm \sqrt {12 ^ 2-4 (-4) (- 9)}} {2 (-4)}\\x = \frac {-12 \pm \sqrt {144-144}} {- 8}\\x = \frac {-12 \pm0} {- 8}\\x = \frac {-12} {- 8}\\x = \frac {3} {2}[/tex]
The intersection with the x axis is[tex](\frac {3} {2}, 0)[/tex]
Answer:
[tex]d = 0[/tex]
The intersection with the x axis is [tex](\frac {3} {2}, 0)[/tex]
