Question

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

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:

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

Here, the coefficients are:
- \(a = 3\)
- \(b = 6\)
- \(c = -42\)

Now we can calculate the discriminant:

\[
D = 6^2 - 4 \cdot 3 \cdot (-42)
\]
\[
D = 36 + 504
\]
\[
D = 540
\]

Now, we analyze the discriminant:
- If \(D > 0\), there are 2 distinct real solutions.
- If \(D = 0\), there is exactly 1 real solution.
- If \(D < 0\), there are no real solutions.

Since \(D = 540\) (which is greater than 0), we conclude that there are **2 distinct real solutions** for the equation \(3x^2 + 6x - 42 = 0\).