Let p, q and r be three distinct sides of PQR. If the roots of the equation x* + 2(p + q+1) x +3k (pq + gr + pr) - 0 are real, then

The problem appears to involve the roots of a quadratic equation with the coefficients depending on the sides \( p, q, r \) of a triangle \( PQR \), and it asks for conditions that ensure the roots are real.

The given quadratic equation is:

\[
x^2 + 2(p + q + 1) x + 3k (pq + qr + pr) = 0
\]

### Step 1: Discriminant for Real Roots
For the roots of a quadratic equation \( ax^2 + bx + c = 0 \) to be real, the discriminant \( \Delta \) must be non-negative. The discriminant is given by:

\[
\Delta = b^2 - 4ac
\]

In this case:
- \( a = 1 \)
- \( b = 2(p + q + 1) \)
- \( c = 3k(pq + qr + pr) \)

Thus, the discriminant becomes:

\[
\Delta = [2(p + q + 1)]^2 - 4 \times 1 \times 3k(pq + qr + pr)
\]

Simplifying the discriminant:

\[
\Delta = 4(p + q + 1)^2 - 12k(pq + qr + pr)
\]

For the roots to be real, this discriminant must be non-negative, i.e.:

\[
4(p + q + 1)^2 - 12k(pq + qr + pr) \geq 0
\]

### Step 2: Condition for Real Roots
Now, we simplify this inequality:

\[
(p + q + 1)^2 \geq 3k(pq + qr + pr)
\]

This is the condition that must be satisfied for the roots of the quadratic equation to be real.

### Conclusion:
The roots of the equation will be real if and only if the above inequality holds. Depending on the specific values of \( p \), \( q \), \( r \), and \( k \), we can analyze further, but the essential condition is that the discriminant must be non-negative.

Would you like a deeper exploration of this condition, or any further details?

### 5 Relative Questions:
1. What is the geometric significance of \( p, q, r \) as the sides of the triangle?
2. How does the value of \( k \) affect the discriminant and thus the reality of the roots?
3. What happens if the roots of the quadratic equation are complex? What can be inferred about the sides of the triangle?
4. Could we generalize this approach to other quadratic equations involving the sides of a triangle?
5. How would this condition change if \( p, q, r \) were not distinct?

### Tip:
Always check the discriminant of a quadratic equation to determine the nature (real or complex) of the roots.

Let p, q and r be three distinct sides of PQR. If the roots of the equation x^2 + 2(p + q + 1) x + 3k (pq + qr + pr) = 0 are real, then what condition must be satisfied?

Explore how to determine the condition for real roots of the quadratic equation x^2 + 2(p + q + 1)x + 3k(pq + qr + pr) = 0, involving the sides of a triangle. This guide covers discriminants, algebraic concepts, and inequalities, suitable for Grades 11-12.

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