Solve : `2x^(2)-3x+5 lt 0 `

Answer

To solve the inequality \( 2x^2 - 3x + 5 < 0 \), we will follow these steps:

### Step 1: Identify the coefficients
In the quadratic expression \( 2x^2 - 3x + 5 \), we identify the coefficients:
- \( a = 2 \)
- \( b = -3 \)
- \( c = 5 \)

### Step 2: Calculate the discriminant
The discriminant \( D \) of a quadratic equation \( ax^2 + bx + c \) is given by the formula:
\[ D = b^2 - 4ac \]
Substituting the values of \( a \), \( b \), and \( c \):
\[ D = (-3)^2 - 4 \cdot 2 \cdot 5 = 9 - 40 = -31 \]

### Step 3: Analyze the discriminant
Since the discriminant \( D = -31 \) is less than 0, this indicates that the quadratic equation has no real roots.

### Step 4: Determine the nature of the quadratic function
When the discriminant is negative, the quadratic function \( 2x^2 - 3x + 5 \) does not intersect the x-axis and is always positive (since the coefficient of \( x^2 \) is positive).

### Step 5: Conclusion
Since the quadratic expression \( 2x^2 - 3x + 5 \) is always greater than 0 for all \( x \), it cannot be less than 0. Therefore, the solution to the inequality \( 2x^2 - 3x + 5 < 0 \) is:
\[ \text{No solution} \]