The sum of two numbers is 8, and the sum of their squares is 34. what is the larger number

The larger number is 5.

Step-by-step explanation:

Let the numbers be represented by x and y.  Then:

x + y = 8   is their sum, and

x² + y² = 34 is the sum of their squares.

Let's solve this system of equations using substitution.  Since x + y = 8,

x = 8 - y.  Subbing this 8 - y for x into the second equation, we get:

(8 - y)² + y² = 34, or

64 - 16 y + y² + y² = 34

Simplifying, we get first: 30 - 16y + 2y² = 0, and then:

2y² - 16y + 30 = 0, and after reducing, y² - 8y + 15 = 0.

This factors into (y - 3)(y - 5) = 0, and so y = 3 and y = 5.

The larger number is 5.