In the following questions three equations are given in variables x. Third equation is equal to the sum of the first two equations. You have to solve the questions based on given information.

a𝑥2+ bx+ 2 = 0𝑏𝑥2-8x + k = 0

2𝑥2−7x +49−13= 0

To solve the problem based on the given information, let's analyze each equation one by one. The problem states that the third equation is equal to the sum of the first two equations.

Given equations:

1. \( a x^2 + b x + 2 = 0 \)
2. \( b x^2 - 8x + k = 0 \)
3. \( 2x^2 - 7x + 49 - 13 = 0 \)

Let's first simplify the third equation:

\[
2x^2 - 7x + 49 - 13 = 0 
\]
\[
2x^2 - 7x + 36 = 0 
\]

This is our third equation, which should be the sum of the first two equations.

### Step 1: Sum of the first two equations

To sum the first two equations, let's add them:

\[
(a x^2 + b x + 2) + (b x^2 - 8x + k) = (a + b)x^2 + (b - 8)x + (2 + k) = 0
\]

According to the problem, this sum is equal to the third equation:

\[
2x^2 - 7x + 36 = 0
\]

Now, let's match the coefficients of this equation with the summed equation:

1. **For the \(x^2\) term:**
   \[
   a + b = 2
   \]

2. **For the \(x\) term:**
   \[
   b - 8 = -7
   \]

3. **For the constant term:**
   \[
   2 + k = 36
   \]

### Step 2: Solve for \(b\), \(a\), and \(k\)

Let's solve these equations one by one.

1. **Solve for \(b\):**
   \[
   b - 8 = -7
   \]
   \[
   b = -7 + 8 = 1
   \]

2. **Solve for \(a\):**
   \[
   a + b = 2
   \]
   Substituting \(b = 1\):
   \[
   a + 1 = 2
   \]
   \[
   a = 2 - 1 = 1
   \]

3. **Solve for \(k\):**
   \[
   2 + k = 36
   \]
   \[
   k = 36 - 2 = 34
   \]

Thus, the values of the constants are:

- \(a = 1\)
- \(b = 1\)
- \(k = 34\)

### Conclusion

The values of \(a\), \(b\), and \(k\) are \(1\), \(1\), and \(34\), respectively. These satisfy the condition that the third equation is the sum of the first two.

Would you like more details or have any questions about this solution?

Here are some related questions:
1. What are the roots of the third equation \(2x^2 - 7x + 36 = 0\)?
2. How would the solution change if the coefficients of the equations were complex numbers?
3. Can you find a similar relationship for cubic equations?
4. What is the discriminant of a quadratic equation and how does it determine the nature of the roots?
5. How would you verify if your solution to simultaneous equations is correct?

**Tip:** Always check your final values by substituting them back into the original equations to ensure consistency.

Learn how to solve simultaneous equations involving quadratic equations with detailed step-by-step solutions. Understand the relationships between variables and how to derive constants such as a, b, and k. Suitable for high school students studying algebra and quadratic equations.

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