If the roots of the equation ` x^2+a^2 =8x + 6a ` are real , then a lies between

Options

A. [2, 9]
B. [-2, 8]
C. [0, 7]
D. [6, 8]

Answer

To solve the problem, we need to analyze the quadratic equation given by \( x^2 + a^2 = 8x + 6a \) and determine the conditions under which its roots are real.

### Step-by-Step Solution:

1. **Rearranging the Equation:**
Start by rearranging the equation into standard quadratic form:
\[ x^2 - 8x + (a^2 - 6a) = 0 \]
Here, we identify the coefficients:
- \( A = 1 \)
- \( B = -8 \)
- \( C = a^2 - 6a \)

2. **Finding the Discriminant:**
For the roots of the quadratic equation to be real, the discriminant must be non-negative:
\[ D = B^2 - 4AC \geq 0 \]
Substituting the values of \( A \), \( B \), and \( C \):
\[ D = (-8)^2 - 4 \cdot 1 \cdot (a^2 - 6a) \geq 0 \]
Simplifying this gives:
\[ 64 - 4(a^2 - 6a) \geq 0 \]

3. **Simplifying the Discriminant Inequality:**
Distributing the \( -4 \):
\[ 64 - 4a^2 + 24a \geq 0 \]
Rearranging the terms:
\[ -4a^2 + 24a + 64 \geq 0 \]
Dividing the entire inequality by \(-4\) (and reversing the inequality sign):
\[ a^2 - 6a - 16 \leq 0 \]

4. **Factoring the Quadratic:**
Now we need to factor the quadratic \( a^2 - 6a - 16 \):
\[ a^2 - 6a - 16 = (a - 8)(a + 2) \leq 0 \]

5. **Finding the Roots:**
The roots of the equation \( (a - 8)(a + 2) = 0 \) are:
\[ a = 8 \quad \text{and} \quad a = -2 \]

6. **Testing Intervals:**
We will test the intervals determined by the roots:
- Interval 1: \( (-\infty, -2) \)
- Interval 2: \( (-2, 8) \)
- Interval 3: \( (8, \infty) \)

For each interval, we check the sign of \( (a - 8)(a + 2) \):
- In \( (-\infty, -2) \): Both factors are negative, so the product is positive.
- In \( (-2, 8) \): \( (a - 8) \) is negative and \( (a + 2) \) is positive, so the product is negative.
- In \( (8, \infty) \): Both factors are positive, so the product is positive.

7. **Conclusion:**
The inequality \( (a - 8)(a + 2) \leq 0 \) holds true in the interval:
\[ a \in [-2, 8] \]

Thus, the values of \( a \) for which the roots of the equation are real lie between \(-2\) and \(8\), inclusive.