poopsiedoodle 2 years ago How might one go about doing this problem? Solve for x: x^2 + 24x + 90 = 0

1. nooce

Factor first :)

2. nooce

What factors of 90, when added together, give you 24?

3. poopsiedoodle

Gimme a second.

4. nooce

5. poopsiedoodle

I don't think any do. 18 and 5 is the closest you can get to 24, which is 23.

6. nooce

Hmm, hold on a sec

7. poopsiedoodle

I will do that.

8. nooce

Well, since it isn't factorable, we have to use the quadratic formula. Have you learned it?

9. poopsiedoodle

Hardly. I've just started on it.

10. nooce

Well, this may take some explaining to do, so hold on for a bit haha.

11. nooce

x^2 + 24x + 90 = 0 Let's break down what this problem means. By plugging in a certain number(s) for x, you get zero. It's pretty much like plotting a graph of x^2 + 24x + 90 and looking for where the graph touches zero. Did I lose you yet?

12. nooce

|dw:1358120873885:dw|

13. poopsiedoodle

Okay. I get it so far.

14. nooce

Alright. So since we have an equation that isn't factorable, we find the zeroes by plugging it into the quadratic formula.

15. nooce

This is the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac} }{ 2a }$ Now, a, b, and c are the numbers in your equation -- $a^2+bx+c$ In this case: $x^2 + 24x + 90$ so, a=1, b= 24, and c=90.

16. poopsiedoodle

Ok, lemme try it now.

17. nooce

Remember, the ± symbol indicates that there are two expressions! (ex. 1 ± 3 is 1 + 3 AND 1 - 3)

18. poopsiedoodle

$\Large0= \frac{ -24 \pm \sqrt{24^{2} -4(90)} }{ 2 }$

19. nooce

Good so far!

20. nooce

except, it's x =

21. poopsiedoodle

Well, since we're already using x in the original problem, how about Y?

22. nooce

But we're trying to solve for x, haha

23. poopsiedoodle

eh, whatever. By the way, I'm having a slight problem with $\sqrt{4(90)}$

24. nooce

That's just $\sqrt{4 • 90} = \sqrt{360}$

25. poopsiedoodle

Ah. $\Large x= \frac{ -24 \pm 24\sqrt{360} }{ 2 }$

26. nooce

No, remember 24^2 is under the radical (Don't take it out!) $\sqrt{24^2 - 360}$

27. poopsiedoodle

$\Large x= \frac{ -24 \pm \sqrt{ 24^{2} (-360)} }{ 2 }$*

28. poopsiedoodle

But, since √24^2 = 24, what's wrong with taking it out?

29. nooce

Because we can't have a negative number under a square root, haha

30. poopsiedoodle

Ah.

31. nooce

But anyways, after plugging all of those numbers in, what do you geT?

32. poopsiedoodle

So, $\Large x= { -12 \pm (\sqrt{ 24^{2} (-360)} /2)}$ ? I'm confused now e_o

33. nooce

Well, the /2 should be on the bottom of the whole thing, but besides that, Let's simplify the square root first. What is 24^2 - 360?

34. poopsiedoodle

216.

35. poopsiedoodle

and the root of that is 14.giganticdecimal

36. nooce

Well, let's simplify it haha sqrt of 216 can be reduced into what? (hint: 36 x 6 = 216)

37. poopsiedoodle

oh gee, this is a tought one... 2... and... erm... GIMME A MINUTE

38. poopsiedoodle

tough*

39. poopsiedoodle

OOH! I KNOW! 36 AND 6. BOOM. Don't you wish you had thought of that?

40. nooce

Hahahaha. You are correct x)

41. nooce

So... now we have $x = \frac{ -24 \pm 6\sqrt{6} }{ 2 }$ which breaks down to: $x = \frac{ -24 + 6\sqrt{6} }{ 2 }$ $x = \frac{ -24 - 6\sqrt{6} }{ 2 }$ I assume you can solve from there :)

42. poopsiedoodle

$\Huge x = { -12 - 3\sqrt{3} }$ ?

43. nooce

Annnnnnnnnd?

44. poopsiedoodle

I was wondering if I was right, which I guess I am. So, is it $$\Huge x=-12 -3?$$

45. nooce

Wait, no don't take out the radical 3 lol You need to solve for the other part: $x = \frac{ -24 + 6\sqrt{6} }{ 2 }$ Your first solution was correct, btw!

46. poopsiedoodle

Oh. x=-12 + 3√3. And now what do I do?

47. nooce

Well, those are your solutions, haha x=-12 + 3√3 and x=-12 - 3√3 Yay!

48. poopsiedoodle

:D

49. nooce

:D indeed! If you need more help, just pm me (: