Use the discriminant to determine the number of real solutions of 3x2+6x−42=0

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 \(3x^2 + 6x - 42 = 0\).