Question

Find the points at which the following curves intersect (correct to 2 decimal places)
y=x^2+3x-10
x^2+y^2-2x-6y-6=0

Answer 1

I'd try it by substituting the first equation into the second and solving for x.

Answer 2

It's possible for a parabola and an ellipse to intersect in, at most, 4 places, so be sure you can handle solving a 4th-degree polynomial.

Answer 3

4th -degree polynomial?

Answer 4

http://www3.wolframalpha.com/Calculate/MSP/MSP25801a1ghh332bg1a54b000055i95f6g142edeh7?MSPStoreType=image/gif&s=13&w=470&h=293&cdf=RangeControl

Answer 5

ive subbed the first equation in the second, then i get stuck there

Answer 6

Yes, it's possible that after doing the substitution that you'll get an equation with x^4

Answer 7

Did you simplify as much as possible?

Answer 8

yeah i think

Answer 9

on the picture there is curve and point

Answer 10

i got \[5x ^{2}+y ^{2}-20x-54=0\]

Answer 11

From the graph ajay posted, you can see that the 4th-degree equation will only have 2 real solutions. Are you familiar with the techniques of using the rational root theorem and synthetic division?

Answer 12

no, not really

Answer 13

When you do the substitution, you should get an equation with only one variable.

Answer 14

so i can sub y into y^2?

Answer 15

It is impossible to solve a single equation that has 2 variables in it.
Start here, and simplify:
\[x^2+(x^2+3x+10)^2-2x-6(x^2+3x+10)-6\]
This is merely equation one substituted into equation two.

Answer 16

Yes, the first equation is solved for y, so you can use that expression for y as a substitute in the second equation.

Answer 17

okay ill try the simplifying part

Answer 18

how do you do x^2* 3x

Answer 19

@CliffSedge
?

Answer 20

The rest is going to be very difficult if you don't know how to multiply monomials.
\[x^2*3x=3x^3\]

Answer 21

I'm going to double check my simplifying, I was expecting some negative terms..

Answer 22

@SirMathy what methods have you been taught to solve this?
e.g. maybe you are being asked to plot these graphs and see where they intersect, or maybe you have been asked to use numerical analysis (like the Newton-Raphson Method)?

Answer 23

http://www.wolframalpha.com/input/?i=y%3Dx^2%2B3x-10+and++x^2%2By^2-2x-6y-6%3D0

Answer 24

From that analysis, it looks like the two real solutions are irrational - that can be difficult to figure out by hand..

Answer 25

my math teacher said to use substitution

Answer 26

in that case, when you substitute the first equation into the second you end up with:\[x^4+6x^3-16x^2-80x+154=0\]have you been taught how to solve such equations?

Answer 27

I asked, and SirMathy says he doesn't know the rational root theorem or synthetic division..

Answer 28

then we are missing something here. are you sure you have the right question?

Answer 29

Yeah, look dont worry, ill ask my math teacher about this question tommorrow
Thankyou for your help anyway

Answer 30

ok - yw