Question 7 (1 point)
Determine the number of roots for the function $f(x)=4 x^{2}-5 x+12$.
A

Step 1 ：The number of roots of a quadratic function can be determined by the discriminant, which is calculated as \(b^{2}-4ac\) where \(a\), \(b\), and \(c\) are the coefficients of the quadratic equation \(ax^{2}+bx+c=0\). If the discriminant is greater than 0, the function has two roots. If the discriminant is equal to 0, the function has one root. If the discriminant is less than 0, the function has no real roots.

Step 2 ：In this case, \(a=4\), \(b=-5\), and \(c=12\). We can substitute these values into the discriminant formula to find the number of roots.

Step 3 ：\(a = 4\)

Step 4 ：\(b = -5\)

Step 5 ：\(c = 12\)

Step 6 ：Calculate the discriminant: \(discriminant = b^{2}-4ac = (-5)^{2}-4*4*12 = -167\)

Step 7 ：Since the discriminant is less than 0, the function has no real roots.

Step 8 ：Final Answer: The function \(f(x)=4 x^{2}-5 x+12\) has \(\boxed{0}\) real roots.