Find a Cartesian equation for the curve. $$ r=5 \cos (\theta) $$ $$ \square $$

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To convert the polar equation $$ r = 5 \cos(\theta) $$ into a Cartesian equation, we can use the relationships between polar and Cartesian coordinates:

$$
x = r \cos(\theta)
$$
$$
y = r \sin(\theta)
$$
$$
r = \sqrt{x^2 + y^2}
$$

From the polar equation $$ r = 5 \cos(\theta) $$, we can multiply both sides by $$ r $$:

$$
r^2 = 5r \cos(\theta)
$$

Now, substituting $$ r^2 $$ with $$ x^2 + y^2 $$ and $$ r \cos(\theta) $$ with $$ x $$:

$$
x^2 + y^2 = 5x
$$

Next, we can rearrange this equation to standard form:

$$
x^2 - 5x + y^2 = 0
$$

To complete the square for the $$ x $$ terms, we take the coefficient of $$ x $$, which is $$-5$$, halve it to get $$-\frac{5}{2}$$, and then square it to get $$\left(-\frac{5}{2}\right)^2 = \frac{25}{4}$$.

Now, we add and subtract $$\frac{25}{4}$$ in the equation:

$$
x^2 - 5x + \frac{25}{4} + y^2 = \frac{25}{4}
$$

This can be rewritten as:

$$
\left(x - \frac{5}{2}\right)^2 + y^2 = \frac{25}{4}
$$

This is the equation of a circle with center $$\left(\frac{5}{2}, 0\right)$$ and radius $$\frac{5}{2}$$.

Thus, the Cartesian equation for the curve is:

$$
\left(x - \frac{5}{2}\right)^2 + y^2 = \frac{25}{4}
$$
The Cartesian equation for the curve is $$\left(x - \frac{5}{2}\right)^2 + y^2 = \frac{25}{4}$$.

Find a Cartesian equation for the curve.

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