To determine the number and type of roots for the quadratic equation \( 2x^2 - 7x + 9 = 0 \), we first need to identify the coefficients \( a \), \( b \), and \( c \).
In this equation:
- \( a = 2 \)
- \( b = -7 \)
- \( c = 9 \)
Next, we'll calculate the discriminant \( D \) using the formula:
\[ D = b^2 - 4ac \]
Substituting the values of \( a \), \( b \), and \( c \):
\[ D = (-7)^2 - 4(2)(9) \] \[ D = 49 - 72 \] \[ D = -23 \]
The discriminant \( D = -23 \) is less than 0.
According to the properties of the discriminant:
- If \( D > 0 \), there are two distinct real roots.
- If \( D = 0 \), there is one real root (a repeated root).
- If \( D < 0 \), there are two complex (non-real) roots.
Since \( D = -23 \) is less than 0, we conclude that the equation \( 2x^2 - 7x + 9 = 0 \) has two complex (non-real) roots.