If `x^2+2ax+10-3a gt0` for all `x in R` then

Options

A. `-5 lt a lt 2`
B. `a lt -5`
C. `a gt 5`
D. `2 lt a lt 5`

Answer

To solve the inequality \( x^2 + 2ax + 10 - 3a > 0 \) for all \( x \in \mathbb{R} \), we need to analyze the quadratic expression.

### Step-by-Step Solution:

1. **Identify the coefficients**:
The given quadratic expression can be rewritten in the standard form \( ax^2 + bx + c \), where:
- \( a = 1 \)
- \( b = 2a \)
- \( c = 10 - 3a \)

2. **Condition for the quadratic to be positive**:
For the quadratic \( ax^2 + bx + c \) to be greater than 0 for all \( x \in \mathbb{R} \), the following conditions must be satisfied:
- The leading coefficient \( a \) must be greater than 0.
- The discriminant \( D \) must be less than 0, where \( D = b^2 - 4ac \).

3. **Check the leading coefficient**:
Since \( a = 1 \), which is greater than 0, this condition is satisfied.

4. **Calculate the discriminant**:
We need to ensure that the discriminant \( D < 0 \):
\[ D = b^2 - 4ac = (2a)^2 - 4 \cdot 1 \cdot (10 - 3a) \]
Simplifying this gives:
\[ D = 4a^2 - 4(10 - 3a) = 4a^2 - 40 + 12a \]
Thus, we have:
\[ D = 4a^2 + 12a - 40 \]

5. **Set up the inequality**:
To ensure the quadratic is always positive, we require:
\[ 4a^2 + 12a - 40 < 0 \]

6. **Divide the entire inequality by 4**:
This simplifies our inequality:
\[ a^2 + 3a - 10 < 0 \]

7. **Factor the quadratic**:
We can factor \( a^2 + 3a - 10 \):
\[ (a + 5)(a - 2) < 0 \]

8. **Determine the intervals**:
The roots of the equation \( (a + 5)(a - 2) = 0 \) are \( a = -5 \) and \( a = 2 \). To find the intervals where the product is negative, we test the intervals:
- For \( a < -5 \): both factors are negative (product is positive).
- For \( -5 < a < 2 \): one factor is positive and one is negative (product is negative).
- For \( a > 2 \): both factors are positive (product is positive).

Therefore, the solution to the inequality is:
\[ -5 < a < 2 \]

### Conclusion:
The values of \( a \) for which \( x^2 + 2ax + 10 - 3a > 0 \) for all \( x \in \mathbb{R} \) are:
\[ a \in (-5, 2) \]