Question

f(x) = 4x^2+2x+6f(x)=4x 2 +2x+6f, left parenthesis, x, right parenthesis, equals, 4, x, squared, plus, 2, x, plus, 6 What is the value of the discriminant of fff? How many distinct real number zeros does fff have?

Answer 1

Given:

The given function is $f(x)=4x^2+2x+6$

We need to determine the value of the discriminant f and also to determine the distinct real number zeros of f.

Discriminant:

The discriminant can be determined using the formula,

$\Delta = b^2-4ac$

Now, we shall determine the discriminant of the function $f(x)=4x^2+2x+6$

Substituting the values in the formula, we have;

$\Delta=(2)^2-4(4)(6)$

$\Delta=4-96$

$\Delta=-92$

Thus, the value of the discriminant of f is -92.

Distinct roots:

The distinct zeros of the function f can be determined by

(1) If $\Delta>0$, then the function has 2 real roots.

(2) If $\Delta=0$, then the function has 2 real roots ( or one repeated root).

(3) If $\Delta$, then the function has 2 imaginary roots (or no real roots).

Since, the discriminant is $\Delta=-92 \ < \ 0$ , then the function has no real roots  or 2 imaginary roots.

Thus, the function has 2 imaginary roots.