How many real solutions does the following quadratic equation have?

Question

4x2+x+3=0
4
𝑥
2
+
𝑥
+
3
=
0

(1 point)
Responses

one real solution
one real solution

no real solutions
no real solutions

three real solutions
three real solutions

two real solutions

Answer 1

To determine the number of real solutions for the quadratic equation \(4x^2 + x + 3 = 0\), we can use the discriminant method. The discriminant \(D\) of a quadratic equation in the form \(ax^2 + bx + c = 0\) is given by:

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

Here, \(a = 4\), \(b = 1\), and \(c = 3\).

We can calculate the discriminant:

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

Since the discriminant is negative (\(D < 0\)), the quadratic equation has no real solutions.

Thus, the correct response is:

no real solutions.