Math Snap

STEP 1

What is this asking?
Where do the parabolas $y = x^2 + 7x - 14$ and $y = -3x + 10$ cross?
Watch out!
Don't forget there might be two crossing points!

STEP 2

1. Set up the equation
2. Solve for $x$
3. Solve for $y$

STEP 3

Alright, so we want to find where these two parabolas meet, which means they'll have the same $x$ and $y$ values at those points.
Since both equations are already solved for $y$, let's set them equal to each other!
This gives us $x^2 + 7x - 14 = -3x + 10$.
This is where the magic happens!

STEP 4

Now, let's rewrite the equation to make it easier to solve.
We want it to look like a standard quadratic equation, which has the form $ax^2 + bx + c = 0$.
So, let's add $3x$ to both sides of our equation.
This gives us $x^2 + 10x - 14 = 10$.
Remember, we're adding $3x$ to both sides to keep the equation balanced!

STEP 5

Next, let's subtract $10$ from both sides to get everything on one side.
This gives us $x^2 + 10x - 24 = 0$.
Awesome! Now we have a nice, clean quadratic equation.

STEP 6

Time to solve for $x$!
We can do this by factoring.
We're looking for two numbers that multiply to $-24$ and add up to $10$.
Those numbers are $12$ and $-2$.
So, we can rewrite our equation as $(x + 12)(x - 2) = 0$.

STEP 7

Now, if $(x + 12)(x - 2) = 0$, then either $x + 12 = 0$ or $x - 2 = 0$.
This gives us two possible solutions for $x$: $x = -12$ and $x = 2$.
Two solutions?
That means our parabolas cross at two points!

STEP 8

We've got our $x$ values, so now we need to find the corresponding $y$ values.
Let's use the simpler equation, $y = -3x + 10$, to do this.

STEP 9

For $x = -12$, we have $y = -3 \cdot (-12) + 10 = 36 + 10 = 46$.
So, one intersection point is $(-12, 46)$.

STEP 10

For $x = 2$, we have $y = -3 \cdot 2 + 10 = -6 + 10 = 4$.
So, the other intersection point is $(2, 4)$.

The graphs intersect at the points $(-12, 46)$ and $(2, 4)$.