Question

if (x = -3) is the only (x)-intercept of the graph of a quadratic equation, which statement best describes the discriminant of the equation?

• the discriminant is negative.
• the discriminant is (-3).
• the discriminant is (0).
• the discriminant is positive.

Explanation:

⚡ Using what you learned: quadratic formula and its applications

Step 1: Relate x-intercepts to real roots

An \(x\)-intercept of a quadratic graph \(y = ax^2 + bx + c\) corresponds to a real solution (or root) of the quadratic equation:

$$ ax^2 + bx + c = 0 $$

Since the problem states that \(x = -3\) is the only \(x\)-intercept, the quadratic equation has exactly one unique real solution.

Step 2: Analyze the discriminant

The number of real solutions of a quadratic equation is determined by its discriminant, \(b^2 - 4ac\):

• If \(b^2 - 4ac > 0\), there are two distinct real solutions (two \(x\)-intercepts).
• If \(b^2 - 4ac = 0\), there is exactly one real solution (one \(x\)-intercept).
• If \(b^2 - 4ac < 0\), there are no real solutions (no \(x\)-intercepts).

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

Answer:

The discriminant is 0.