Question

the graph of an equation with a negative discriminant always has which characteristic?

no \\(x\\)-intercept
no \\(y\\)-intercept
no maximum
no minimum

Explanation:

⚡ Using what you learned: quadratic formula and its applications

Step 1: Understand the discriminant

The discriminant of a quadratic equation \(ax^2 + bx + c = 0\) is given by:

$$ D = b^2 - 4ac $$

A negative discriminant (\(D < 0\)) means the quadratic equation has no real solutions (only complex solutions).

Step 2: Relate solutions to the graph

The real solutions of the equation \(ax^2 + bx + c = 0\) correspond to the \(x\)-intercepts of the graph of the quadratic function \(y = ax^2 + bx + c\).

Since there are no real solutions when the discriminant is negative, the graph never crosses or touches the \(x\)-axis. Therefore, the graph has no \(x\)-intercept.

Answer:

no x-intercept