Find the value of a and b in x² + 16x + a + b²

To find the values of a and b in the expression x² + 16x + a + b², we start by rewriting the quadratic part.

The expression can be grouped as follows:

x² + 16x

Next, we can complete the square for the quadratic term:

1. Take the coefficient of x, which is 16.

2. Divide it by 2, giving us 8.

3. Square it, resulting in 64.

Now we can rewrite the expression using this completed square:

(x + 8)² - 64

We need to adjust our original expression with the addition of a and b²:

(x + 8)² - 64 + a + b²

For the expression to remain as a perfect square, we want:

-64 + a + b² = 0

Rearranging gives:

a + b² = 64

This equation shows the relationship between a and b, but without further information, we cannot determine unique values for a and b. However, we know they must satisfy this equation where the sum of a and b² equals 64.

For example, if we let b = 0, then:

a + 0² = 64

This makes a = 64. Likewise, if b = 8, we get:

a + 8² = 64

Which simplifies to a = 0. Therefore, there are many possible pairs of (a, b) such as (64, 0) or (0, 8) that satisfy the equation.