To find the value of $p$ for which the quadratic equation

$(p+1) x^2 - 6(p+1) x + 3(p+9) = 0$

has equal roots, we need to use the discriminant condition for a quadratic equation.

Step 1: Standard form of a quadratic equation

A quadratic equation is generally written as:

$Ax^2 + Bx + C = 0$

where $A$, $B$, and $C$ are constants. For the given equation, we can identify:

• $A = p + 1$
• $B = -6(p + 1)$
• $C = 3(p + 9)$

Step 2: Condition for equal roots

For a quadratic equation to have equal roots, the discriminant must be zero. The discriminant $\Delta$ is given by:

$\Delta = B^2 - 4AC$

For equal roots, we require $\Delta = 0$.

Substituting the values of $A$, $B$, and $C$:

$\Delta = \left[-6(p + 1)\right]^2 - 4(p + 1)(3(p + 9))$

Step 3: Simplify the discriminant

Let's first expand both terms:

1. The square of $B$: $\left[-6(p + 1)\right]^2 = 36(p + 1)^2$

2. The second term involving $A$ and $C$: $4(p + 1) \cdot 3(p + 9) = 12(p + 1)(p + 9)$

Now, equate the discriminant to zero:

$36(p + 1)^2 - 12(p + 1)(p + 9) = 0$

Step 4: Factor out the common term

Both terms have $(p + 1)$ as a common factor, so factor it out:

$(p + 1)\left[36(p + 1) - 12(p + 9)\right] = 0$

This gives us two cases to consider:

Case 1: $p + 1 = 0$

This gives:

$p = -1$

Case 2: The second factor equals zero

Now, solve the equation inside the brackets:

$36(p + 1) - 12(p + 9) = 0$

Simplify this expression:

$36p + 36 - 12p - 108 = 0$ $24p - 72 = 0$ $24p = 72$ $p = 3$

Step 5: Check the roots for both values of $p$

Now, we check the roots for both values of $p$.

When $p = -1$:

Substitute $p = -1$ into the original quadratic equation:

$(0)x^2 - 6(0)x + 3(8) = 0$

This simplifies to:

$24 = 0$

This is a contradiction, so $p = -1$ is not a valid solution.

When $p = 3$:

Substitute $p = 3$ into the original equation:

$(3 + 1)x^2 - 6(3 + 1)x + 3(3 + 9) = 0$ $4x^2 - 24x + 36 = 0$

Divide the entire equation by 4:

$x^2 - 6x + 9 = 0$

This is a perfect square trinomial, so we can factor it as:

$(x - 3)^2 = 0$

Thus, the roots are $x = 3$, and they are equal.

Conclusion:

The value of $p$ for which the quadratic equation has equal roots is $p = 3$, and the equal root is $x = 3$.

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Would you like more details on any of the steps, or do you have any other questions?

Here are some related questions for further exploration:

1. How do you find the discriminant of a quadratic equation?
2. What does the discriminant tell us about the nature of the roots?
3. How can you factor a quadratic equation like $x^2 - 6x + 9 = 0$?
4. What happens if the discriminant is positive, negative, or zero?
5. How do you solve quadratic equations by completing the square?

Tip: Always remember that the discriminant $\Delta = B^2 - 4AC$ helps determine the number and type of roots in a quadratic equation. If $\Delta = 0$, the quadratic equation has equal roots.