Question

How many real solutions does the following quadratic equation have?

4x2+x+3=0

(1 point)
Responses

two real solutions
two real solutions

three real solutions
three real solutions

one real solution
one real solution

no real solutions
no real solutions
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Answer 1

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

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

For the quadratic equation \(ax^2 + bx + c = 0\), we have:
- \(a = 4\)
- \(b = 1\)
- \(c = 3\)

Now, we calculate the discriminant:

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

Since the discriminant \(D\) is negative (\(-47 < 0\)), this indicates that the quadratic equation has **no real solutions**.

Thus, the correct answer is:

**no real solutions**.