To find the value of $p$ for which the quadratic equation $(p+6)x^2 + (p+3)x + 1 = 0$ has equal roots, we use the condition for equal roots of a quadratic equation, which is when the discriminant ($\Delta$) is zero.

For a general quadratic equation $ax^2 + bx + c = 0$, the discriminant is given by:

$\Delta = b^2 - 4ac$

Here, we have:

• $a = p+6$
• $b = p+3$
• $c = 1$

Now, apply these values to the discriminant formula:

$\Delta = (p+3)^2 - 4(p+6)(1)$

Setting $\Delta = 0$ for equal roots:

$(p+3)^2 - 4(p+6) = 0$

Expanding both terms:

$(p+3)^2 = p^2 + 6p + 9$ $4(p+6) = 4p + 24$

Substituting these into the equation:

$p^2 + 6p + 9 - 4p - 24 = 0$

Simplify:

$p^2 + 2p - 15 = 0$

Now, solve this quadratic equation:

$p^2 + 2p - 15 = 0$

Factoring the equation:

$(p + 5)(p - 3) = 0$

Thus, the solutions are:

$p = -5 \quad \text{or} \quad p = 3$

So, the values of $p$ for which the quadratic equation has equal roots are $p = -5$ or $p = 3$.

Would you like further explanation or additional details on this? Here are some related questions to explore:

1. What are the properties of the discriminant for quadratic equations?
2. How do we determine if a quadratic equation has distinct, real, or complex roots?
3. Can you explain the process of factoring a quadratic equation?
4. What is the geometric interpretation of equal roots for a parabola?
5. How does the discriminant affect the nature of roots?

Tip: Always check the discriminant first when solving quadratic equations to know the nature of the roots.