Is this the correct formula for this quadratic equation?

Question

I'm doing some excersises, but i'm not sure who to apply the 'formula' given:

$$x^2 - 2px + p^2 - 1 = 0$$.

I've found this formula on my book:

Is it the correct 'formula'?

If it were something like:

$$x^2 -2x + 9 - 1 = 0$$ i would make it, but the "p" confuses me. Any hint is welcome.

Answer 1

The quadratic formula will work fine if you apply it and simplify. However in this case you might like to use the idea that any quadratic is of the form $$x^2-(\text{sum of roots})x+(\text{product of roots})=0$$ In this case the product of roots is $(p+1)(p-1)$

Answer 2

Comparing $$a x^2 + b x + c =0$$ with $$x^2 - 2p \, x + p^2 - 1 = 0$$

you should see that, in your case, $a=1$, $b=-2p$ and $c= p^2-1$

Answer 3

That is the right formula, you just set your coefficients appropriately. So $a=1$, $b=-2p$, and $c=p^2-1$. Can you proceed from here?

Answer 4

$$ \left[x^2 - 2px + p^2\right] - 1 = 0 \\ (x - p)^2 - (1)^2 = 0 \\ \left[(x - p) - 1)\right]\left[(x - p) + 1\right] = 0$$

Giving you $ x = p \pm 1 $