The image contains a mathematical problem that asks to find two numbers whose sum is 13 and the sum of their squares is 85.

To solve this, we can set up two equations based on the information provided:

1. Let the two numbers be $x$ and $y$.
2. According to the problem: $x + y = 13$ $x^2 + y^2 = 85$

Step-by-step solution:

1. Express $y$ in terms of $x$: $y = 13 - x$

2. Substitute $y$ in the second equation: $x^2 + (13 - x)^2 = 85$

3. Expand and simplify: $x^2 + (169 - 26x + x^2) = 85$ $2x^2 - 26x + 169 = 85$

4. Move all terms to one side: $2x^2 - 26x + 169 - 85 = 0$ $2x^2 - 26x + 84 = 0$

5. Divide the entire equation by 2 to simplify: $x^2 - 13x + 42 = 0$

6. Factor the quadratic equation: $(x - 7)(x - 6) = 0$

7. Solve for $x$: $x = 7 \quad \text{or} \quad x = 6$

8. Find corresponding $y$ values:

   • If $x = 7$, then $y = 13 - 7 = 6$
   • If $x = 6$, then $y = 13 - 6 = 7$

Thus, the two numbers are 6 and 7.

Would you like more details or have any other questions?

Related Questions to Explore:

1. How can we solve similar problems using substitution methods?
2. What is the quadratic formula and how is it used in finding the roots of a quadratic equation?
3. How does factoring help in solving quadratic equations?
4. What are the properties of sums and differences of squares?
5. How can we use graphs to solve simultaneous equations?

Tip:

When dealing with word problems, always start by defining variables and writing down equations based on the given information. This helps organize your solution methodically.