Question

find two functions defined implicitly by this equation.
$x^2 - y^2 = 25$
$y = \pm \sqrt{?x^2 + }$

Explanation:

Step1: Rearrange the given equation

We start with the equation \(x^{2}-y^{2}=25\). Let's solve for \(y^{2}\) first. Add \(y^{2}\) to both sides and subtract 25 from both sides:
\(x^{2}- 25=y^{2}\)
Which can be rewritten as \(y^{2}=x^{2}-25\)

Step2: Solve for y

Now, to solve for \(y\), we take the square root of both sides. Remember that when we take the square root of a variable squared, we get both the positive and negative roots. So:
\(y=\pm\sqrt{x^{2}-25}\)
Comparing with the given form \(y = \pm\sqrt{[?]x^{2}+[ ]}\), we can see that the coefficient of \(x^{2}\) is \(1\) and the constant term is \(- 25\) (or we can think of it as \(x^{2}-25=1\times x^{2}+(- 25)\))

Answer:

The coefficient in the green box (for \(x^{2}\)) is \(1\) and the constant in the gray box is \(-25\), so the functions are \(y=\pm\sqrt{1x^{2}+(-25)}\) (or simplified as \(y = \pm\sqrt{x^{2}-25}\))