Question

3. Having survived the meteor impact, thanks to some last-minute evasive maneuvers, the Mathonauts now set their sights on their Interstellar Headquarters. The Interstellar Headquarters orbits the Earth based on the equation y2 + x2 = 40,000. Using the original trajectory of the ship and complete sentences, explain to the pilot how to find where the ship’s path will cross the Interstellar Headquarters’ path.

The equation I chose was y=2x+5 and I need to know how to plug it into y and solve for x.

Answer 1

you have \[\Large y^2 + x^2 = 40,000\]

Answer 2

replace y with 2x+5. This is valid because y = 2x+5

\[\Large y^2 + x^2 = 40,000\]

\[\Large (2x+5)^2 + x^2 = 40,000\]

Answer 3

I know I need to get x to one side and solve

Answer 4

then you expand out (2x+5)^2

\[\Large (2x+5)^2 + x^2 = 40,000\]

\[\Large (2x+5)(2x+5) + x^2 = 40,000\]

\[\Large 2x(2x+5)+5(2x+5) + x^2 = 40,000\]

\[\Large 4x^2+10x+10x+25 + x^2 = 40,000\]

Answer 5

I'll let you finish up

Answer 6

Do I have to foil the x^2 so I can get x by itself and to one side of the equation?

Answer 7

I recommend combining like terms for the next step

Answer 8

I was thinking of that but I have to get x to one side of the equation because I am trying to solve for x. I have to find two points so I can put them on a graph when I am done.

Answer 9

you'll need to use the quadratic formula

Answer 10

but first you need to get everything to one side

Answer 11

could you help me with that? I just need help with plugging in my equation for y and getting x to one side of the equation and then I can solve the rest on my own.

Answer 12

\[\Large 4x^2+10x+10x+25 + x^2 = 40,000\]
\[\Large 4x^2+10x+10x+25 + x^2-40,000 = 0\]
\[\Large 5x^2+20x-39,975 = 0\]

Answer 13

what's the next step?

Answer 14

What happened to the y?

Answer 15

I replaced it with 2x+5

Answer 16

since y = 2x+5

Answer 17

I did that up near the top

Answer 18

Im not sure from here because x isnt on one side

Answer 19

do you know the quadratic formula?

Answer 20

ax2 + bx + c = 0

Answer 21

that's part of it

Answer 22

what was that you posted?

Answer 23

i was trying to post the second part but it came out funny. it was x=-b plus or negative square root b^2 -4ac/2a

Answer 24

I really need help Im not sure how to solve it until only x is on one side and not 0.

Answer 25

I gotcha, so this

\[\Large x = \frac{-b\pm \sqrt{b^2-4ac}}{2a}\]

Answer 26

what are a,b,c in this case?

Answer 27

a=5 b=20 and c= -39,975

Answer 28

so you'll plug those values into the formula I wrote above

Answer 29

Do I keep the negative on 39,975

Answer 30

yes you do, c is negative

Answer 31

This is what I came up with.
x=87.4371287553441
x=−91.4371287553441

Answer 32

let me check

Answer 33

I'm getting the same. Those are the approximate solutions

Answer 34

Is there something easier we could do? because this is too much work for a simple word problem. I only need one x solution and one y solution. It should be where I plug the equation for y move x to one side of the equation by subtracting or something and then solve and whatever my answer is I plug it back in for y and solve. I need something easy.

Answer 35

with quadratics, you could either factor or use the quadratic formula or complete the square

the quadratic formula works with any quadratic while factoring only works with some quadratics

Answer 36

I know that but this is not like my homework lesson. in the lesson we solved by substitution or elimination.

Answer 37

It was way easier than using the quadratic formula

Answer 38

well in that first step I used substitution to replace y with 2x+5

Answer 39

after plugging in the equation for y could we just factor out x and then subtract x to one side?

Answer 40

that method won't work unfortunately

Answer 41

oh... well then what am I supposed to do with two x solutions?

Answer 42

Im only supposed to have one x and one y solutions to plot on a graph

Answer 43

the answer at the end of all this is supposed to be where these two equations intersect

Answer 44

does it say that x has to be positive?

Answer 45

no but at this point i think it is supposed to be positive

Answer 46

You can substitute your original trajectory ( y=5x+1 ) into the equation that the Interstellar Headquarters is based on ( y2+x2=40,000 ). You will be with the equation 2( 5x+1 )+x2 = 40,000 and you will solve for X. After solving for X, you will have X=3333.12 for your value. You will now take this value back into one of your original equations. You will now have the equation y=5 ( 3333.17 )+1 and will solve for Y. Your value for Y will be Y=16666.85. You now know the ship’s path will cross the Interstellar Headquarters’ at the coordinate ( 3333.17, 16666.85 ).

Answer 47

This is from someone with the same question as me and they made it real simple just ignore all the question marks

Answer 48

they made an error

Answer 49

they thought the y^2 was 2y

Answer 50

maybe it is supposed to be that because if its y^2 and x^2 then we would come up with more than one x solution and i can only have one.

Answer 51

well usually y2 is often y^2 because I've seen where the symbols don't paste properly

Answer 52

could we try solving it the same way that person did and see what we come up with?

Answer 53

I solved their equation and I got 2 solutions also

Answer 54

not just 1 solution

Answer 55

so I'm not sure which solution they want

Answer 56

maybe if we solved for y we might come up with 2 solutions as well

Answer 57

yeah I think that will happen as well