Given the cubic equation $x^3 - 9x^2 + ax - 27 = 0$, and the fact that it has equal and positive real roots, we are tasked with finding the value of $a$.

Step 1: Understanding the roots

Let the equation have three equal roots, denoted by $r$. Since all the roots are equal and positive, we can express the cubic equation as: $(x - r)(x - r)(x - r) = 0$ This simplifies to: $(x - r)^3 = 0$ Expanding the cubic term: $x^3 - 3r x^2 + 3r^2 x - r^3 = 0$ By comparing the expanded form of the equation $x^3 - 3r x^2 + 3r^2 x - r^3 = 0$ with the given equation $x^3 - 9x^2 + ax - 27 = 0$, we can match the coefficients.

Step 2: Matching coefficients

From the comparison, we get the following system of equations:

• Coefficient of $x^2$: $-3r = -9$, which implies $r = 3$.
• Coefficient of $x$: $3r^2 = a$, which gives $a = 3(3)^2 = 27$.
• Constant term: $-r^3 = -27$, which confirms $r^3 = 27$ and thus $r = 3$, consistent with our earlier result.

Final answer

The value of $a$ is $\boxed{27}$.

Do you want more details or have any questions?

Here are 5 related questions:

1. How would the solution change if the roots were not equal?
2. What if the roots were equal but not positive? How would this affect the equation?
3. Can you derive a formula to solve cubic equations with equal roots?
4. What other properties can you infer about a cubic polynomial with equal roots?
5. How do the sum and product of roots relate to the coefficients in a cubic equation?

Tip: For cubic equations with equal roots, the coefficient of the $x^2$ term is always proportional to the root value multiplied by 3.