ANSWER
The quadratic function,
[tex]y = 12 {x}^{2} - 9x + 4[/tex]
has no real roots.
EXPLANATION
We use the discriminant,
[tex]D = {b}^{2} - 4ac[/tex]
to determine the nature of the roots of a quadratic equation.
If we compare
[tex]y = 12 {x}^{2} - 9x + 4[/tex]
to
[tex]y = a{x}^{2} + bx + c[/tex]
Then
[tex]a=12,b=-9,c=4[/tex]
We substitute these values in to the formula for the discriminant, to obtain,
[tex]D = {( - 9)}^{2} - 4(12)(4)[/tex]
[tex]D = 81 - 192[/tex]
This implies that,
[tex]D = - 111 \: < \: 0[/tex]
Since the discriminant is zero, the quadratic function,
[tex]y = 12 {x}^{2} - 9x + 4[/tex]
has no real roots.
