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Let and be the two solutions from the quadratic formula.
Finish this identity: . Are there any restrictions on what can be for the identity to be true?
Pause here for a discussion.
Mathematical identities can be useful in a variety of situations.
In an earlier course, we learned how to derive the quadratic formula using the method of completing the square on a quadratic equation. We can check the formula by reconstructing the standard form of a quadratic equation using the zeros guaranteed by the quadratic formula. We can start with writing the factored form using the zeros as .
After distributing the left side of the equation, it nicely becomes .
Identities can also be used to rewrite expressions to be in a form that is clearer to understand. For example, the value of the fraction is not very easy to estimate without a calculator. Let’s rewrite the expression using two identities: the difference of squares identity, , and the identity when .
In this form, we can estimate that is approximately 2.5 and is a little more than 1, so the expression has a value of approximately 1.25. Checking on a calculator, we can see that the original fraction and the final expression are equal and have a value of approximately 1.232, which is close to our estimation.