Given that 𝑥² = (√(13) − √(11))(√(13) + √(11)), find the possible values of 𝑥.

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### Video Transcript

Given that 𝑥 squared is equal to the square root of 13 minus the square root of 11 times the square root of 13 plus the square root of 11, find the possible values of 𝑥.

If we know that 𝑥 squared equals the square root of 13 minus the square root of 11 times the square root of 13 plus the square root of 11, we might recognize the form 𝑎 minus 𝑏 times 𝑎 plus 𝑏. And we recognize that this form is the difference of squares. 𝑎 minus 𝑏 times 𝑎 plus 𝑏 is equal to 𝑎 squared minus 𝑏 squared. Under these conditions, we’ll let 𝑎 equal the square root of 13 and 𝑏 equal the square root of 11. And that means we can say 𝑥 squared equals the square root of 13 squared minus the square root of 11 squared. The square root of 13 squared equals 13. The square root of 11 squared equals 11. And that means 𝑥 squared equals 13 minus 11. 𝑥 squared equals two.

If we want to know all possible values of 𝑥, we can take the square root of both sides of the equation. And that will mean there will be a positive and a negative solution of the square root of two. If 𝑥 squared equals two, then negative square root of two squared would equal two. And positive square root of two squared would equal two. And so, we say that 𝑥 could be the positive square root of two or the negative square root of two. Solving using this method depends on you recognizing the form for difference of squares. But what if you didn’t remember that? Could you solve this question without knowing the difference of squares form?

We could do that, but we would need to expand this multiplication. We would need to multiply the square root of 13 by the square root of 13. Then we would multiply the square root of 13 by the square root of 11. From there, we would multiply the negative square root of 11 times the square root of 13. And we would get the negative square root of 13 times the square root of 11. And then we would have the negative square root of 11 times the positive square root of 11. The square root of 13 times the square root of 13 is the square root of 13 squared, and that equals 13. The same thing for the square root of 11 times the square root of 11. It equals the square root of 11 squared, which is 11.

We have to be careful here because we were dealing with negative square root of 11 times positive square root of 11. That means we’ll have negative 11. What about these two middle terms? We have positive square root of 13 times the square root of 11 minus square root of 13 times the square root of 11. These values cancel each other out. And so, we’re left with 𝑥 squared is equal to 13 minus 11, which is just what we found with the other form. And by taking the square root of both sides, we get 𝑥 equals the square root of two or the negative square root of two.