To determine the number of real solutions of the quadratic equation \( 4x^2 - 3x + 2 = 0 \), we can use the discriminant. The discriminant, denoted as \( D \), is given by the formula:
\[ D = b^2 - 4ac \]
where \( a \), \( b \), and \( c \) are the coefficients of the quadratic equation \( ax^2 + bx + c = 0 \).
In the equation \( 4x^2 - 3x + 2 = 0 \), we have:
- \( a = 4 \)
- \( b = -3 \)
- \( c = 2 \)
Now, we can substitute these values into the discriminant formula:
\[ D = (-3)^2 - 4(4)(2) \]
Calculating \( D \):
\[ D = 9 - 32 \] \[ D = -23 \]
The value of the discriminant is \( -23 \).
Since the discriminant is negative (\( D < 0 \)), this indicates that the quadratic equation has no real solutions. Therefore, there are two complex solutions.