Question

if \\(x = 6\\) is the only \\(x\\)-intercept of the graph of a quadratic equation, which statement best describes the discriminant of the equation?

• the discriminant is 0.
• the discriminant is 6.
• the discriminant is positive.
• the discriminant is negative.

Explanation:

⚡ Using what you learned: quadratic formula and its applications

Step 1: Analyze the number of intercepts

The problem states that \( x = 6 \) is the only \( x \)-intercept of the quadratic function's graph. This means the quadratic equation has exactly one real solution (a double root).

Step 2: Relate to the discriminant

The discriminant of a quadratic equation, \( D = b^2 - 4ac \), determines the number and type of solutions:

• If \( D > 0 \), there are two distinct real solutions (two \( x \)-intercepts).
• If \( D = 0 \), there is exactly one real solution (one \( x \)-intercept).
• If \( D < 0 \), there are no real solutions (no \( x \)-intercepts).

Since there is only one \( x \)-intercept, the discriminant must be \( 0 \).

Answer:

The discriminant is 0.