# Math Help - Graph Intersection

1. ## Graph Intersection

Considering the graphs of y = 3x + c and y^2 = 6x, where c is a real constant, determine all values of c for which the graphs intersect in two distinct points.

I'm not entirely sure of where to start so any help is greatly appreciated.

2. Hello, Naples!

Considering the graphs of: . $\begin{Bmatrix}y \:=\: 3x + c \\ y^2 \:=\: 6x\end{Bmatrix}$ .where $c$ is a real constant,

determine all values of c for which the graphs intersect in two distinct points.
First, find the intersections . . .

The first equation gives us: . $x \:=\:\frac{y-c}{3}$

Substitute into the second: . $y^2 \:=\:6\left(\frac{y-c}{3}\right) \quad\Rightarrow\quad y^2 - 2y + 2c \:=\:0$

Quadratic Formula: . $y \;=\;\frac{2\pm\sqrt{4-8c}}{2}$

The quadratic has two roots if the discriminant is positive: . $4-8c \:>\:0$

Therefore: . $-8c \:>\:-4 \quad\Rightarrow\quad\boxed{ c \:<\:\tfrac{1}{2}}$