Thread: why always?

When I do problems involving real roots, such as x = 1, why is it always x = 1 or x = - 1?

For example:

Determine the real roots of each equation:


the real root is x = 4 or x = - 4. Why is it always +/- or how can I identify this issue?

Originally Posted by Barthayn

When I do problems involving real roots, such as x = 1, why is it always x = 1 or x = - 1?

For example:
Determine the real roots of each equation:

the real root is x = 4 or x = - 4. Why is it always +/- or how can I identify this issue?

But of course that is not always true for all roots. It is true for square roots.
There are two real square roots of 16,
There are no real square roots of

There is only one real cube root of 8, .
There are no others and we did not use .

So basically if you can just square root it it will be +/- always? If it is cubic you can only have one root?

Originally Posted by Barthayn

So basically if you can just square root it it will be +/- always? If it is cubic you can only have one root?

It is very dangerous to answer a question that contains 'always'.
But yes that is generally true for real numbers.

it is because you generated a factor of the form a^2 - b^2 = (a + b)(a - b), the difference of two squares. And why not? They are just exercises, training wheels for a rigorous future math scrimages, if you mind. And multiplicity of roots often comes out handy with some other textbooks . . . .