Question

Use the discriminant to determine the number of real solutions of 3x2+6x−42=0
(1 point)
Responses

two solutions
two solutions

infinite solutions
infinite solutions

one solution
one solution

no solution

Answer 1

To determine the number of real solutions for the quadratic equation \(3x^2 + 6x - 42 = 0\), we can use the discriminant, which is given by the formula:

\[
D = b^2 - 4ac
\]

In the equation \(ax^2 + bx + c = 0\), we have:
- \(a = 3\)
- \(b = 6\)
- \(c = -42\)

Now, we can substitute these values into the discriminant formula:

\[
D = 6^2 - 4 \cdot 3 \cdot (-42)
\]

Calculating \(D\):

\[
D = 36 - 4 \cdot 3 \cdot (-42)
\]
\[
D = 36 + 504
\]
\[
D = 540
\]

Since the discriminant \(D\) is greater than 0 (\(D > 0\)), this indicates that there are **two real solutions** to the quadratic equation.

Thus, the correct response is:
**two solutions**.