Question

At which points are the equations y=x^2+3x+2 and y=2x+3 approximately equal?

Answer 1

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:

$y=x^2+3x+2$

$y=2x+3$.

I'm going to do this by substitution:

$x^2+3x+2=2x+3$

Subtract 2x and 3 on both sides:

$x^2+1x-1=0$

$x^2+x-1=0$

To solve this equation I'm going to use the quadratic formula:

$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

To find $a,b,\text{ and }c$, you must compare $x^2+x-1=0$

to $ax^2+bx+c=0$.

$a=1,b=1,c=-1$.

Now inputting the values into the quadratic formula gives us:

$x=\frac{-1\pm\sqrt{(1)^2-4(1)(-1)}}{2(1)}$

$x=\frac{-1\pm\sqrt{1+4}}{2}$

$x=\frac{-1\pm\sqrt{5}}{2}$

This means you have two solutions:

$x=\frac{-1+\sqrt{5}}{2} \text{ or } x=\frac{-1-\sqrt{5}}{2}$

It does say approximately.

So I'm going to put both of these in my calculator and I guess round to the nearest thousandths.

$x=0.618 \text{ or } x=-1.618$

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).