Video Transcript
Given that 𝑥 over 16 plus the square root of two is equal to 16 minus the square root of two, find the value of 𝑥, stating your answer in simplest form.
In this question, we are given an equation involving an unknown 𝑥, and we want to solve this equation for 𝑥. To solve this equation for 𝑥, we want to isolate 𝑥 on one side of the equation. We can do this by multiplying both sides of the equation by 16 plus the square root of two, where we note that this is nonzero. This then gives us that 𝑥 is equal to 16 minus the square root of two multiplied by 16 plus the square root of two.
We could now distribute the product over the parentheses. However, it is easier to note that this product is in the factored form of a difference of squares. That is, 𝑎 minus 𝑏 multiplied by 𝑎 plus 𝑏 is equal to 𝑎 squared minus 𝑏 squared. Therefore, if we set 𝑎 equal to 16 and 𝑏 equal to the square root of two, we can use the difference between squares formula to rewrite the right-hand side of the equation. We obtain that 𝑥 is equal to 16 squared minus the square root of two squared.
We can then calculate that 16 squared is equal to 256. And we can recall that for any nonnegative real number 𝑎, the square root of 𝑎 all squared is equal to 𝑎. So, we can calculate that the square root of two all squared is equal to two. We can then calculate that 256 minus two is equal to 254.
Hence, if 𝑥 over 16 plus the square root of two is equal to 16 minus root two, then 𝑥 is equal to 254.
