Math Snap

STEP 1

Assumptions1. The first equation is $y = x -3$
. The second equation is y =3x^
3. We are looking for the points of intersection, which are the points $(x, y)$ where both equations are true at the same time.

STEP 2

To find the points of intersection, we can set the two equations equal to each other and solve for $x$.
$$x - =x^2$$

STEP 3

Rearrange the equation to form a quadratic equation.
$$3x^2 - x +3 =0$$

STEP 4

This is a quadratic equation in the form of $ax^2 + bx + c =0$. We can solve it using the quadratic formula, which is $x = \frac{-b \pm \sqrt{b^2 -4ac}}{2a}$.

STEP 5

Plug in the values for $a$, $b$, and $c$ into the quadratic formula.
$$x = \frac{-(-1) \pm \sqrt{(-1)^2 -4<em>3</em>3}}{2*3}$$

STEP 6

implify the equation.
$$x = \frac{1 \pm \sqrt{1 -36}}{6}$$

STEP 7

Calculate the value under the square root.
$$x = \frac{1 \pm \sqrt{-35}}{6}$$

The value under the square root is negative, which means there are no real solutions for $x$. Therefore, the two graphs do not intersect at any real points.