The given quadratic equation is:

$x^2 + 3px + p = 0$

We are told that the equation has equal roots, and $p$ is a non-zero constant. To find the value of $p$, we will use the condition for equal roots in a quadratic equation.

Step 1: Discriminant Condition for Equal Roots

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

$\Delta = b^2 - 4ac$

If the equation has equal roots, the discriminant must be zero, i.e.,

$\Delta = 0$

For the equation $x^2 + 3px + p = 0$, we have:

• $a = 1$
• $b = 3p$
• $c = p$

Step 2: Compute the Discriminant

The discriminant is:

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

Simplifying this expression:

$\Delta = 9p^2 - 4p$

Step 3: Set the Discriminant to Zero

Since the equation has equal roots, we set $\Delta = 0$:

$9p^2 - 4p = 0$

Step 4: Solve for $p$

We can factor this equation as:

$p(9p - 4) = 0$

This gives two possible solutions:

1. $p = 0$
2. $9p - 4 = 0$, which simplifies to $p = \frac{4}{9}$

However, the problem specifies that $p$ is non-zero, so we discard $p = 0$. Thus, the solution is:

$p = \frac{4}{9}$

The value of $p$ is $\frac{4}{9}$.

Would you like more details, or do you have any other questions?

Here are 5 related questions:

1. What are the conditions for distinct roots of a quadratic equation?
2. How does the discriminant help classify the nature of roots?
3. What is the general form of a quadratic equation?
4. How can you factor quadratic equations like the one solved here?
5. What does it mean when a quadratic equation has complex roots?

Tip: Always check for special conditions (like non-zero values) when solving quadratic equations!