Answer:
(0.618,4.236) and (-1.618,-0.236)
Step-by-step explanation:
To find the intersection, we are looking for a common point between the curves.
We are solving the system:
![y=x^2+3x+2](https://tex.z-dn.net/?f=y%3Dx%5E2%2B3x%2B2)
.
I'm going to do this by substitution:
![x^2+3x+2=2x+3](https://tex.z-dn.net/?f=x%5E2%2B3x%2B2%3D2x%2B3)
Subtract 2x and 3 on both sides:
![x^2+1x-1=0](https://tex.z-dn.net/?f=x%5E2%2B1x-1%3D0)
![x^2+x-1=0](https://tex.z-dn.net/?f=x%5E2%2Bx-1%3D0)
To solve this equation I'm going to use the quadratic formula:
![x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}](https://tex.z-dn.net/?f=x%3D%5Cfrac%7B-b%5Cpm%5Csqrt%7Bb%5E2-4ac%7D%7D%7B2a%7D)
To find
, you must compare
to
.
.
Now inputting the values into the quadratic formula gives us:
![x=\frac{-1\pm\sqrt{(1)^2-4(1)(-1)}}{2(1)}](https://tex.z-dn.net/?f=x%3D%5Cfrac%7B-1%5Cpm%5Csqrt%7B%281%29%5E2-4%281%29%28-1%29%7D%7D%7B2%281%29%7D)
![x=\frac{-1\pm\sqrt{1+4}}{2}](https://tex.z-dn.net/?f=x%3D%5Cfrac%7B-1%5Cpm%5Csqrt%7B1%2B4%7D%7D%7B2%7D)
![x=\frac{-1\pm\sqrt{5}}{2}](https://tex.z-dn.net/?f=x%3D%5Cfrac%7B-1%5Cpm%5Csqrt%7B5%7D%7D%7B2%7D)
This means you have two solutions:
![x=\frac{-1+\sqrt{5}}{2} \text{ or } x=\frac{-1-\sqrt{5}}{2}](https://tex.z-dn.net/?f=x%3D%5Cfrac%7B-1%2B%5Csqrt%7B5%7D%7D%7B2%7D%20%5Ctext%7B%20or%20%7D%20x%3D%5Cfrac%7B-1-%5Csqrt%7B5%7D%7D%7B2%7D)
It does say approximately.
So I'm going to put both of these in my calculator and I guess round to the nearest thousandths.
![x=0.618 \text{ or } x=-1.618](https://tex.z-dn.net/?f=x%3D0.618%20%5Ctext%7B%20or%20%7D%20x%3D-1.618)
Now to find the corresponding y coordinates, I need to use one the equations along with each x.
I choose the linear equation: y=2x+3.
y=2x+3 when x=0.618
y=2(0.618)+3=4.236
So one approximate point is (0.618,4.236).
y=2x+3 when x=-1.618
y=2(-1.618)+3=-0.236
So another approximate point is (-1.618,-0.236).