To determine the number of real solutions for the quadratic equation \(4x^2 + x + 3 = 0\), we can use the discriminant (\(D\)) from the quadratic formula.
The discriminant is given by the formula:
\[ D = b^2 - 4ac \]
where \(a\), \(b\), and \(c\) are the coefficients from the quadratic equation \(ax^2 + bx + c = 0\).
For our equation:
- \(a = 4\)
- \(b = 1\)
- \(c = 3\)
Now, we can calculate the discriminant:
\[ D = (1)^2 - 4(4)(3) \] \[ D = 1 - 48 \] \[ D = -47 \]
Since the discriminant \(D\) is less than 0, it indicates that there are no real solutions to the equation \(4x^2 + x + 3 = 0\).
Thus, the answer is:
no real solutions.
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