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