How many real solutions does the following quadratic equation have?

4x^2+x+3=0

(1 point)
Responses

two real solutions
two real solutions
no real solutions
no real solutions
three real solutions
three real solutions
one real solution

1 answer

To determine the number of real solutions to the quadratic equation \( 4x^2 + x + 3 = 0 \), we can use the discriminant, which is given by the formula:

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

For the given quadratic equation, the coefficients are:

  • \( a = 4 \)
  • \( b = 1 \)
  • \( c = 3 \)

Now, we can calculate the discriminant:

\[ D = 1^2 - 4 \cdot 4 \cdot 3 = 1 - 48 = -47 \]

Since the discriminant \( D \) is negative, the quadratic equation has no real solutions.

Therefore, the correct response is:

no real solutions