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The equation given is \(x^2 - 8x + k = 0\), and it asks to find \(k\), given that the roots \(x_1\) and \(x_2\) satisfy the equation \(3x_1 + 4x_2 = 29\).

### Step 1: Use the properties of quadratic equations

For a quadratic equation of the form:
\[
x^2 - (x_1 + x_2)x + x_1x_2 = 0
\]
The coefficients relate to the roots \(x_1\) and \(x_2\) as follows:
- The sum of the roots: \(x_1 + x_2 = 8\) (since the coefficient of \(x\) is \(-8\)).
- The product of the roots: \(x_1 \cdot x_2 = k\) (since the constant term is \(k\)).

### Step 2: Use the condition \(3x_1 + 4x_2 = 29\)

We are given the equation \(3x_1 + 4x_2 = 29\). We already know that \(x_1 + x_2 = 8\). Let's use this information to find \(x_1\) and \(x_2\).

We can solve this system of equations:
1. \(x_1 + x_2 = 8\)
2. \(3x_1 + 4x_2 = 29\)

From the first equation, we can express \(x_1\) as:
\[
x_1 = 8 - x_2
\]

### Step 3: Substitute into the second equation

Substituting \(x_1 = 8 - x_2\) into the second equation:
\[
3(8 - x_2) + 4x_2 = 29
\]
Simplifying:
\[
24 - 3x_2 + 4x_2 = 29
\]
\[
24 + x_2 = 29
\]
\[
x_2 = 5
\]

### Step 4: Find \(x_1\)

Now that we know \(x_2 = 5\), substitute it back into \(x_1 + x_2 = 8\):
\[
x_1 + 5 = 8 \implies x_1 = 3
\]

### Step 5: Find \(k\)

Since \(x_1 \cdot x_2 = k\), we have:
\[
k = 3 \cdot 5 = 15
\]

Thus, the value of \(k\) is \( \boxed{15} \).

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Would you like more details on any of the steps? Here are some additional questions that expand on this problem:

1. How can we use Vieta’s formulas to solve problems involving quadratic roots?
2. What is the geometric interpretation of the sum and product of the roots of a quadratic equation?
3. Can you generalize this method to find \(k\) if the coefficients of the linear combination in the second equation change?
4. How does the discriminant help in determining the nature of the roots in quadratic equations?
5. What happens if the sum or product of the roots has a different relationship with the coefficients?

**Tip**: In problems involving quadratic equations and conditions on roots, always check if Vieta's formulas can simplify the relations between the roots and coefficients.

Given that the roots of the equation x^2 - 8x + k = 0 satisfy 3x1 + 4x2 = 29, find k.

Learn how to find the constant k in the quadratic equation x^2 - 8x + k = 0 by solving for the roots that satisfy 3x1 + 4x2 = 29. This example covers Vieta's formulas and quadratic equations, suitable for students in Grade 9-10.

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