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

Answer

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

### Step 1: Rearranging the Inequality
First, we rearrange the inequality to a standard form:
\[ -2x^2 + 3x - 10 < 0 \]
This can also be written as:
\[ 2x^2 - 3x + 10 > 0 \]

### Step 2: Identifying Coefficients
In the quadratic expression \(2x^2 - 3x + 10\), we identify the coefficients:
- \(a = 2\)
- \(b = -3\)
- \(c = 10\)

### Step 3: Calculating the Discriminant
Next, we calculate the discriminant \(D\) using the formula:
\[ D = b^2 - 4ac \]
Substituting the values we found:
\[ D = (-3)^2 - 4 \cdot 2 \cdot 10 = 9 - 80 = -71 \]

### Step 4: Analyzing the Discriminant
Since the discriminant \(D < 0\), this indicates that the quadratic equation \(2x^2 - 3x + 10 = 0\) has no real roots. Therefore, the parabola does not intersect the x-axis.

### Step 5: Determining the Direction of the Parabola
Since the coefficient of \(x^2\) (which is \(a = 2\)) is positive, the parabola opens upwards. Because it does not intersect the x-axis and opens upwards, the entire parabola is above the x-axis.

### Step 6: Conclusion
Since \(2x^2 - 3x + 10 > 0\) for all real numbers \(x\), we conclude that:
\[ 3x - 10 - 2x^2 < 0 \quad \text{for all } x \in \mathbb{R} \]

### Final Answer
The solution to the inequality \(3x - 10 - 2x^2 < 0\) is:
\[ x \in \mathbb{R} \]

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