Question

f(x) = x² - 3x + 18
what is the value of the discriminant of f?

how many distinct real number zeros does f have?

Explanation:

Part 1: Find the discriminant

Step1: Recall discriminant formula

For a quadratic function \( f(x) = ax^2 + bx + c \), the discriminant \( D \) is given by \( D = b^2 - 4ac \).

Step2: Identify \( a \), \( b \), \( c \)

In \( f(x) = x^2 - 3x + 18 \), we have \( a = 1 \), \( b = -3 \), \( c = 18 \).

Step3: Calculate discriminant

Substitute into the formula: \( D = (-3)^2 - 4(1)(18) \)
\( = 9 - 72 \)
\( = -63 \)

Step1: Recall discriminant rules

• If \( D > 0 \), two distinct real zeros.
• If \( D = 0 \), one real zero (a repeated root).
• If \( D < 0 \), no real zeros.

Step2: Analyze discriminant value

We found \( D = -63 \), which is less than 0.

Answer:

-63

Part 2: Number of distinct real zeros