2x2-5x+9<0

To solve the inequality \( 2x^2 - 5x + 9 < 0 \), we first need to determine if the quadratic equation \( 2x^2 - 5x + 9 = 0 \) has any real roots. We can do this by calculating the discriminant (\( D \)) of the quadratic equation, which is given by the formula:

\[ D = b^2 - 4ac \]

For the equation \( 2x^2 - 5x + 9 \):

• \( a = 2 \)
• \( b = -5 \)
• \( c = 9 \)

Calculating the discriminant:

\[ D = (-5)^2 - 4 \cdot 2 \cdot 9 \] \[ D = 25 - 72 \] \[ D = -47 \]

Since the discriminant is negative (\( D < 0 \)), this means that the quadratic equation has no real roots.

Now, since the leading coefficient (coefficient of \( x^2 \), which is \( 2 \)) is positive, the parabola opens upwards. Since the quadratic does not intersect the x-axis and opens upwards, it will always be above the x-axis for all real \( x \).

Thus, \( 2x^2 - 5x + 9 \) is always greater than 0 for all values of \( x \).

Therefore, the original inequality \( 2x^2 - 5x + 9 < 0 \) has no solutions.