To determine how many real solutions the quadratic equation \( 4x^2 + x + 3 = 0 \) has, we can use the discriminant method. The discriminant \( D \) of a quadratic equation \( ax^2 + bx + c = 0 \) is given by:
\[ D = b^2 - 4ac \]
For the given equation, we identify the coefficients:
- \( a = 4 \)
- \( b = 1 \)
- \( c = 3 \)
Now, we can calculate the discriminant:
\[ D = (1)^2 - 4(4)(3) = 1 - 48 = -47 \]
Since the discriminant \( D \) is negative (\( -47 < 0 \)), this indicates that the quadratic equation has no real solutions.
Thus, the correct response is:
no real solutions.
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