# How do you know if x^2-10x-y+18=0 is a hyperbola, parabola, circle or ellipse?

Aug 7, 2018

This is a parabola.

#### Explanation:

Given:

${x}^{2} - 10 x - y + 18 = 0$

Note that the only term of degree $> 1$ is ${x}^{2}$.

Since the multiplier of $y$ is non-zero, we can deduce that this equation represents a parabola.

In fact, adding $y$ to both sides and transposing, it becomes:

$y = {x}^{2} - 10 x + 18$

which clearly expresses $y$ as a quadratic function of $x$, and hence a parabola with vertical axis.

We can also complete the square to find:

$y = {\left(x - 5\right)}^{2} - 7$

allowing us to identify the vertex $\left(5 , 7\right)$ and axis $x = 5$.

graph{(x^2-10x-y+18)(x-5+0.0001y)((x-5)^2+(y+7)^2-0.01) = 0 [-6.455, 13.545, -7.88, 2.12]}