Find the discriminant of the quadratic equation. Determine the number of real solutions.

Question

3x^2 +12x+12=0
What is the discriminant of the quadratic equation?
How many real solutions does the quadratic equation have?
1)two real solutions
2)one real solution
3)no real solutions

Answer 1

To find the discriminant of the quadratic equation in the form \( ax^2 + bx + c = 0 \), we use the formula for the discriminant:

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

For the quadratic equation \( 3x^2 + 12x + 12 = 0 \), we identify the coefficients as follows:

• \( a = 3 \)
• \( b = 12 \)
• \( c = 12 \)

Now we can plug these values into the discriminant formula:

\[ D = (12)^2 - 4(3)(12) \]

Calculating each part:

\[ D = 144 - 144 = 0 \]

Since the discriminant \( D = 0 \), we can determine the number of real solutions.

In general:

• If \( D > 0 \), there are two distinct real solutions.
• If \( D = 0 \), there is exactly one real solution (a repeated or double root).
• If \( D < 0 \), there are no real solutions.

Since \( D = 0 \), this quadratic equation has:

One real solution.

Therefore, the answer regarding the number of real solutions is:

2. one real solution.