### Video Transcript

Given that 𝑥 is greater than or equal to negative 10 and less than or equal to negative one and 𝑦 is greater than or equal to one and less than or equal to nine, find the smallest possible value of 𝑥 squared plus 𝑦 squared.

We know that when squaring any positive or negative number, we will always get a positive answer. In this question, 𝑥 contains only negative values. Therefore, 𝑥 squared will always be positive. In a similar way, 𝑦 contains only positive values. Therefore, 𝑦 squared will always be positive.

We know that 𝑥 must be greater than or equal to negative 10 and less than or equal to negative one. This means that the maximum value of 𝑥 squared will be equal to negative 10 squared. Negative 10 multiplied by negative 10 is equal to 100. The minimum value of 𝑥 squared will be equal to negative one squared. Negative one multiplied by negative one is equal to one. We can therefore conclude that 𝑥 squared must be greater than or equal to one and less than or equal to 100.

We can now repeat this process for the variable 𝑦. We were told that 𝑦 is greater than or equal to one and less than or equal to nine. The maximum value of 𝑦 squared is therefore equal to nine squared. This is equal to 81. The minimum value of 𝑦 squared is equal to one squared, which equals one. We can therefore conclude that 𝑦 squared must be greater than or equal to one and less than or equal to 81.

We now need to calculate the smallest possible value of 𝑥 squared plus 𝑦 squared. As we are calculating the sum of 𝑥 squared and 𝑦 squared, we need both of these values to be as small as possible. The minimum value of 𝑥 squared plus 𝑦 squared is therefore equal to one plus one. This gives us an answer of two.

If 𝑥 is greater than or equal to negative 10 and less than or equal to negative one and 𝑦 is greater than or equal to one and less than or equal to nine, then the smallest possible value of 𝑥 squared plus 𝑦 squared is two. This occurs when 𝑥 is equal to negative one and 𝑦 is equal to one.