The sum of two numbers is 6. The sum of the squares of the same two numbers is 26. What are the numbers?

Steps to Solve

1. Define the Variables

Let the two numbers be $x$ and $y$. According to the problem, we have two equations:

• The first equation from the sum of the numbers: $$ x + y = 6 $$
• The second equation from the sum of their squares: $$ x^2 + y^2 = 26 $$

2. Express One Variable in Terms of the Other

From the first equation, we can express $y$ in terms of $x$: $$ y = 6 - x $$

3. Substitute into the Second Equation

Now, substitute $y = 6 - x$ into the second equation: $$ x^2 + (6 - x)^2 = 26 $$

4. Expand and Simplify

Expand the equation: $$ x^2 + (36 - 12x + x^2) = 26 $$

Combine like terms: $$ 2x^2 - 12x + 36 = 26 $$

5. Set Up a Quadratic Equation

Rearranging gives us: $$ 2x^2 - 12x + 10 = 0 $$

6. Divide the Equation by 2

To simplify, divide the entire equation by 2: $$ x^2 - 6x + 5 = 0 $$

7. Factor the Quadratic

Now, factor the quadratic: $$ (x - 5)(x - 1) = 0 $$

8. Find the Values of x

Set each factor equal to zero:

• $x - 5 = 0 \Rightarrow x = 5$
• $x - 1 = 0 \Rightarrow x = 1$

9. Find Corresponding y Values

Using $y = 6 - x$:

• If $x = 5$, then $y = 1$.
• If $x = 1$, then $y = 5$.

The pairs of $(x, y)$ are (5, 1) and (1, 5).