Help please :c
Ralph wants to find the solution to a system of equations using y = x - 2 and y = x2 + 6x - 38. Joey says that Ralph can solve x2 + 5x - 36 = 0 to find the x coordinates of the solutions to his system. Explain and demonstrate why Joey is correct.

Question

Help please :c
Ralph wants to find the solution to a system of equations using y = x - 2 and y = x2 + 6x - 38. Joey says that Ralph can solve x2 + 5x - 36 = 0 to find the x coordinates of the solutions to his system. Explain and demonstrate why Joey is correct.

anonymous · Answer

@zepdrix any idea?

zepdrix · Answer

So what Joey did was realize that the `solutions` to the system are where they equal one another,$$\Large\rm x-2=x^2+6x-38$$

zepdrix · Answer

Understand how to proceed from there?

anonymous · Answer

Yes, you would add 2 correct?

zepdrix · Answer

add 2, subtract x,
to get the 0 on the left.

anonymous · Answer

x+6-36=0?

anonymous · Answer

x-30=0

zepdrix · Answer

what? +_+

anonymous · Answer

x=30?

anonymous · Answer

I suck at math I'm sorry :c

zepdrix · Answer

Subtracting x from each side, x-x removes the x on the left side,$$\Large\rm -2=x^2+6x-38-x$$Adding 2 to each side, -2+2 cancel out as well on the left,$$\Large\rm 0=x^2+6x-38-x+2$$Then combine like terms,
6x-x
-38+2

anonymous · Answer

0+x^2+6x^2-36?

anonymous · Answer

0=***

zepdrix · Answer

No you didn't combine the x terms.

anonymous · Answer

Would I add or subtract the x from 6x?

zepdrix · Answer

subtract..

anonymous · Answer

Pshh I knew that.. 0=x^2+6-36

zepdrix · Answer

No no no.
You don't combine like terms like that.

If x is an apple
Then we have 6apples minus 1apple.
Leaves us with 5apples ( or 5 x's ).

6x-x = 5x

anonymous · Answer

Oh.. Like I said I'm terrible at math..

anonymous · Answer

So 0=x^2+5x-36

zepdrix · Answer

$$\Large\rm 0=x^2+5x-36$$Mmm ok good.
And if we solve this, we can show that Joey's method in fact works.

anonymous · Answer

Okay, so we'd solve this by using the quadratic formula right?

zepdrix · Answer

We could, but this one factors quite nicely.

zepdrix · Answer

You want values that `multiply` to -36,
but also `add` to positive 5.

zepdrix · Answer

Let's try a few combinations.

$$\Large\rm -6\cdot 6=36,\qquad\qquad -6+6\ne 5$$Hmm those factors didn't work.

zepdrix · Answer

Hmm this combination is also no good.
$$\Large\rm -18\cdot2=36,\qquad\qquad -18+2\ne5$$

zepdrix · Answer

Woops those should say -36 in the examples*

anonymous · Answer

12 and 3?

zepdrix · Answer

12-3 = 9
Hmm those don't give us 5 either.
Can you think of any other factors of 39? :o

Think of your 9's table.

zepdrix · Answer

of 36*

anonymous · Answer

9 and 4?

anonymous · Answer

9*4=36

zepdrix · Answer

Mmm ok let's try those.
In order that we get a -36, one of them needs to be negative.

You want the 9 or 4 to be negative.

anonymous · Answer

The 4 :oo

zepdrix · Answer

Mmm$$\Large\rm -4\cdot9=-36,\qquad\qquad -4+9=5\qquad \color{green}{\checkmark}$$

zepdrix · Answer

Ok great we found our factors!

anonymous · Answer

so you'd end up with 0=(x+9)(x-4)

anonymous · Answer

Oh okay. So x=9 and x=-4? :oo

anonymous · Answer

Or would it be switched around so 9 would be negative and 4 would be positive?

zepdrix · Answer

I dunno why I said it would be subtraction.
You had it right the first time -_-
Sorry bout that, I must be tired...

0=(x+9)(x-4)

anonymous · Answer

okay.. So my answer would be x=-9 and x=4 correct? And don't worry about it, putting up with my stupidity would make me tired too xD

zepdrix · Answer

Mmmm yah good job \c:/

anonymous · Answer

Thank you so much! c: