Let be a complex number such that Find the largest possible value of

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Let $z$ be a complex number such that $|z| = 1.$ Find the largest possible value of $|z^2 + z - 1|.$

Guest Mar 22, 2022

proyaop · Answer 1 · 2022-03-24T10:53:24

If |z| = 1, then z could be 1 and -1.

We can see there are multiple possible values of z.

First plugging in 1 to the quadratic, we end up with |1 + 1 - 1| = 1

Plugging in -1 to the quadratic, we end up with |1 - 1 - 1| = 1

As you can see, the only value of |z^2 + z - 1| is 1.

Thus,

The largest possible value of $|z^2 + z - 1|$ is 1.