Given that `alpha and beta` are the roots of the equation `x^(2)=7x+4`,
(i) show that `alpha^(3)=53alpha+28`
(ii) find the value of `(alpha)/(beta)+(beta)/(alpha)`.

Answer

To solve the given problem step by step, we will address both parts of the question sequentially.

### Given:
The equation is \( x^2 = 7x + 4 \).

### (i) Show that \( \alpha^3 = 53\alpha + 28 \)

**Step 1: Rewrite the equation in standard form.**
We can rewrite the equation as:
\[ x^2 - 7x - 4 = 0 \]

**Step 2: Use the fact that \( \alpha \) is a root of the equation.**
Since \( \alpha \) is a root, we have:
\[ \alpha^2 - 7\alpha - 4 = 0 \]
From this, we can express \( \alpha^2 \) in terms of \( \alpha \):
\[ \alpha^2 = 7\alpha + 4 \]

**Step 3: Multiply both sides by \( \alpha \) to find \( \alpha^3 \).**
Now, multiply the equation \( \alpha^2 = 7\alpha + 4 \) by \( \alpha \):
\[ \alpha^3 = \alpha \cdot \alpha^2 = \alpha(7\alpha + 4) \]
Expanding this gives:
\[ \alpha^3 = 7\alpha^2 + 4\alpha \]

**Step 4: Substitute \( \alpha^2 \) back into the equation.**
Now, substitute \( \alpha^2 \) from Step 2 into the equation:
\[ \alpha^3 = 7(7\alpha + 4) + 4\alpha \]
This simplifies to:
\[ \alpha^3 = 49\alpha + 28 + 4\alpha \]
Combining like terms results in:
\[ \alpha^3 = 53\alpha + 28 \]

Thus, we have shown that:
\[ \alpha^3 = 53\alpha + 28 \]