Question

find two consecutive odd integers such that the sum of their squares is 394.

Explanation:

Define the variables

Using the Consecutive Integer Word Problems knowledge point
Let \(n\) be the smaller odd integer.
The next consecutive odd integer is \(n + 2\).

Set up the equation

Using the Consecutive Integer Word Problems knowledge point

$$ n^2 + (n + 2)^2 = 394 $$

Expand and simplify the equation

Using the Square of a Binomial Sum knowledge point

$$
LATEXBLOCK0
$$

Solve the quadratic equation

Using the Solving Quadratic Equations knowledge point

$$
LATEXBLOCK1
$$

Determine the integer pairs

Using the Consecutive Integer Word Problems knowledge point
If \(n = 13\), the consecutive odd integers are \(13\) and \(15\).
If \(n = -15\), the consecutive odd integers are \(-15\) and \(-13\).
Since standard consecutive integer problems typically seek positive integer solutions unless specified otherwise, we select the positive pair.
Smaller integer: \(13\)
Larger integer: \(15\)

Answer:

Find two consecutive odd integers such that the sum of their squares is 394.

smaller integer = <blank>13</blank>

larger integer = <blank>15</blank>