To determine the number of real solutions of 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:
\[ D = b^2 - 4ac \]
For the equation \(4x^2 + x + 3 = 0\):
- \(a = 4\)
- \(b = 1\)
- \(c = 3\)
Now, we calculate the discriminant:
\[ D = (1)^2 - 4(4)(3) = 1 - 48 = -47 \]
Since the discriminant \(D\) is less than 0, this means that the quadratic equation has no real solutions.
So the correct response is:
no real solutions
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