Question Video: Evaluating Algebraic Expressions Involving Algebraic Identities Mathematics

If (π‘₯ + 𝑦)Β² = 100 and π‘₯𝑦 = 20, what is the value of π‘₯Β² + 𝑦²?

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Video Transcript

If π‘₯ plus 𝑦 all squared is equal to 100 and π‘₯𝑦 equals 20, what is the value of π‘₯ squared plus 𝑦 squared?

So, we’ve been given two pieces of information about these numbers π‘₯ and 𝑦 and asked to use them to determine the value of π‘₯ squared plus 𝑦 squared. Now, your first thought may be that π‘₯ plus 𝑦 all squared is just equal to π‘₯ squared plus 𝑦 squared. In which case, the value we’re looking for is the value given in the question; it’s 100. But if this is the case, why have we also been given the value of π‘₯𝑦?

In fact, if we were to answer the question this way, we’d have made one of the most common mistakes in mathematics because we’ve incorrectly expanded the binomial. Remember that π‘₯ plus 𝑦 all squared means π‘₯ plus 𝑦 multiplied by π‘₯ plus 𝑦. So, we are, in fact, multiplying a binomial by itself, not just squaring each of the individual terms.

Let’s see what happens if we correctly expand π‘₯ plus 𝑦 all squared. Using the FOIL method which, remember, stands for firsts, outers, inners, lasts, this gives π‘₯ squared plus π‘₯𝑦 plus π‘₯𝑦 plus 𝑦 squared, which simplifies to π‘₯ squared plus two π‘₯𝑦 plus 𝑦 squared. What we now have is an equation connecting π‘₯ plus 𝑦 all squared, whose value we know, π‘₯𝑦, whose value we know, and π‘₯ squared plus 𝑦 squared, whose value we wish to calculate.

Substituting 100 for π‘₯ plus 𝑦 all squared and 20 for π‘₯𝑦, we have 100 equals π‘₯ squared plus 𝑦 squared plus two multiplied by 20. That simplifies to 100 equals π‘₯ squared plus 𝑦 squared plus 40. And subtracting 40 from each side, we find that π‘₯ squared plus 𝑦 squared is equal to 60. So, we’ve solved the problem. By correctly expanding the binomial π‘₯ plus 𝑦 all squared and then substituting the values given in the question, we found that π‘₯ squared plus 𝑦 squared is equal to 60.

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