Question

analyzing equations use the discriminant to match the quadratic equation ( 3x^{2}+6x - 9 = 0 ) with the correct graph.
the discriminant is
this means that the graph has
x - intercepts. the y - intercept is

Explanation:

Step1: Find the discriminant

For a quadratic equation \(ax^{2}+bx + c = 0\), the discriminant is \(\Delta=b^{2}-4ac\).
Given \(3x^{2}+6x - 9=0\), where \(a = 3\), \(b = 6\), \(c=-9\).

$$
LATEXBLOCK0
$$

Step2: Determine the number of \(x\) - intercepts

Since \(\Delta>0\), the quadratic equation has two distinct real roots. So the graph has two \(x\) - intercepts.

Step3: Find the \(y\) - intercept

To find the \(y\) - intercept, set \(x = 0\) in the equation \(y=3x^{2}+6x - 9\).
When \(x = 0\), \(y=-9\).

Answer:

The discriminant is \(144\). This means that the graph has two \(x\) - intercepts. The \(y\) - intercept is \(-9\).