Step by step
The question
Question: solve y = −x² + 4 and y = x + 2.
We call them ① and ②.
We want the points (x, y) that make both true.
① is a parabola, and ② is a line.
Not both lines, so this is called a nonlinear system.
1 Put ② into ①
In ②, y is already alone: y = x + 2.
Both ① and ② say what y is.
So the two right sides are equal.
2 Get 0 on one side
We take x + 2 away from both sides.
Careful!
The answers are points, not just x numbers.
After x, we must find each y.
Now we have −x² − x + 2 = 0.
Every number can be divided by −1.
We divide both sides by −1.
0 divided by −1 is still 0.
3 Factor
We need two numbers.
They multiply to −2, and they add to 1.
They multiply to a minus number, so one number is plus and one is minus.
They add to a plus number, so the bigger one is plus.
We list the pairs that make 2, smallest first.
We take the smaller from the bigger.
We need 1.
1 and 2 give 1.
Yes!
With the signs: −1 and 2.
So it is (x − 1)(x + 2) = 0.
4 One bracket is 0
Two numbers multiply to 0.
That only happens if one of them is 0.
So one bracket must be 0.
We make each bracket 0.
Two small equations.
In x − 1 = 0, we add 1 on both sides.
In x + 2 = 0, we take away 2 on both sides.
So x = 1 or x = −2.
5 Find y
Each x goes into the line ②: x = −2 gives y = 0.
And x = 1 gives y = 3.
So the two points are (−2, 0) and (1, 3).
6 Check
Let's check with a picture.
We draw both on a grid.
① is a parabola.
② is a line.
They meet at (−2, 0) and (1, 3): our answers.
Let's check each point in both equations.
First (−2, 0) in ①: the right side is 0, the same as y.
In ②: 0 again.
It works!
Now (1, 3) in ①: the right side is 3, the same as y.
In ②: 3 again.
It works!
Let's look back.
First: put ② into ①.
Then: get 0 on one side.
Then: factor.
Then: one bracket is 0.
Then: find y.
Last: check.