Solve : `3x-10-2x^(2) lt 0`

Answer

To solve the inequality \(3x - 10 - 2x^2 < 0\), we will follow these steps:

### Step 1: Rearrange the inequality
First, we can rearrange the inequality to bring all terms to one side:
\[ -2x^2 + 3x - 10 < 0 \]

### Step 2: Factor out -1
To make the leading coefficient positive, we can factor out -1:
\[ -(2x^2 - 3x + 10) < 0 \]
This simplifies to:
\[ 2x^2 - 3x + 10 > 0 \]

### Step 3: Determine the discriminant
Next, we need to find the discriminant of the quadratic equation \(2x^2 - 3x + 10\) to analyze its roots. The discriminant \(D\) is given by:
\[ D = b^2 - 4ac \]
where \(a = 2\), \(b = -3\), and \(c = 10\). Plugging in these values:
\[ D = (-3)^2 - 4 \cdot 2 \cdot 10 = 9 - 80 = -71 \]

### Step 4: Analyze the discriminant
Since the discriminant \(D = -71\) is less than zero, it indicates that the quadratic equation \(2x^2 - 3x + 10\) has no real roots. Therefore, the quadratic expression \(2x^2 - 3x + 10\) does not cross the x-axis.

### Step 5: Determine the sign of the quadratic
Since the leading coefficient (2) is positive and the quadratic has no real roots, it means that the quadratic expression \(2x^2 - 3x + 10\) is always positive for all real values of \(x\).

### Conclusion
Thus, the solution to the inequality \(3x - 10 - 2x^2 < 0\) is:
\[ \text{All } x \in (-\infty, \infty) \]