SOLUTION: I have a test tomorrow and am having a lot of trouble with quadratic equations. i understand some things, but others do not make sense at all. below are 3 problems that i do not un

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equation Customizable Word Problems -> SOLUTION: I have a test tomorrow and am having a lot of trouble with quadratic equations. i understand some things, but others do not make sense at all. below are 3 problems that i do not un      Log On


   

Question 115684: I have a test tomorrow and am having a lot of trouble with quadratic equations. i understand some things, but others do not make sense at all. below are 3 problems that i do not understand how to solve. if you could solve the problems and show the steps, that would really help me a lot.
1. 3x^2 + 9 = 0
2. 2x^2 - 14x - 10 = 0
3. -2u^2 + 6 = 3u^2 - 10u


i attempted problem one and the answer i ended up with was x = the square root of 3. i thought for those types of problems there were supposed to be two answers though, like x + sq. root three and x equals (another number)
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
#1
Well you're forgetting the other part of the answer which is x=-sqrt%283%29. You are correct, there should be two answers.




#2

Let's use the quadratic formula to solve for x:


Starting with the general quadratic

ax%5E2%2Bbx%2Bc=0

the general solution using the quadratic equation is:

x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29



So lets solve 2%2Ax%5E2-14%2Ax-10=0 ( notice a=2, b=-14, and c=-10)




x+=+%28--14+%2B-+sqrt%28+%28-14%29%5E2-4%2A2%2A-10+%29%29%2F%282%2A2%29 Plug in a=2, b=-14, and c=-10



x+=+%2814+%2B-+sqrt%28+%28-14%29%5E2-4%2A2%2A-10+%29%29%2F%282%2A2%29 Negate -14 to get 14



x+=+%2814+%2B-+sqrt%28+196-4%2A2%2A-10+%29%29%2F%282%2A2%29 Square -14 to get 196 (note: remember when you square -14, you must square the negative as well. This is because %28-14%29%5E2=-14%2A-14=196.)



x+=+%2814+%2B-+sqrt%28+196%2B80+%29%29%2F%282%2A2%29 Multiply -4%2A-10%2A2 to get 80



x+=+%2814+%2B-+sqrt%28+276+%29%29%2F%282%2A2%29 Combine like terms in the radicand (everything under the square root)



x+=+%2814+%2B-+2%2Asqrt%2869%29%29%2F%282%2A2%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)



x+=+%2814+%2B-+2%2Asqrt%2869%29%29%2F4 Multiply 2 and 2 to get 4

So now the expression breaks down into two parts

x+=+%2814+%2B+2%2Asqrt%2869%29%29%2F4 or x+=+%2814+-+2%2Asqrt%2869%29%29%2F4


Now break up the fraction


x=%2B14%2F4%2B2%2Asqrt%2869%29%2F4 or x=%2B14%2F4-2%2Asqrt%2869%29%2F4


Simplify


x=7+%2F+2%2Bsqrt%2869%29%2F2 or x=7+%2F+2-sqrt%2869%29%2F2


So these expressions approximate to

x=7.65331193145904 or x=-0.653311931459037


So our solutions are:
x=7.65331193145904 or x=-0.653311931459037

Notice when we graph 2%2Ax%5E2-14%2Ax-10, we get:



when we use the root finder feature on a calculator, we find that x=7.65331193145904 and x=-0.653311931459037.So this verifies our answer




-2u%5E2%2B6=3u%5E2-10u Start with the given equation


-2u%5E2%2B6-3u%5E2%2B10u=0 Move all of the terms to the left side


-2u%5E2-3u%5E2%2B10u%2B6=0 Sort the terms


-5u%5E2%2B10u%2B6=0 Combine like terms



Let's use the quadratic formula to solve for u:


Starting with the general quadratic

au%5E2%2Bbu%2Bc=0

the general solution using the quadratic equation is:

u+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29



So lets solve -5%2Au%5E2%2B10%2Au%2B6=0 ( notice a=-5, b=10, and c=6)




u+=+%28-10+%2B-+sqrt%28+%2810%29%5E2-4%2A-5%2A6+%29%29%2F%282%2A-5%29 Plug in a=-5, b=10, and c=6



u+=+%28-10+%2B-+sqrt%28+100-4%2A-5%2A6+%29%29%2F%282%2A-5%29 Square 10 to get 100



u+=+%28-10+%2B-+sqrt%28+100%2B120+%29%29%2F%282%2A-5%29 Multiply -4%2A6%2A-5 to get 120



u+=+%28-10+%2B-+sqrt%28+220+%29%29%2F%282%2A-5%29 Combine like terms in the radicand (everything under the square root)



u+=+%28-10+%2B-+2%2Asqrt%2855%29%29%2F%282%2A-5%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)



u+=+%28-10+%2B-+2%2Asqrt%2855%29%29%2F-10 Multiply 2 and -5 to get -10

So now the expression breaks down into two parts

u+=+%28-10+%2B+2%2Asqrt%2855%29%29%2F-10 or u+=+%28-10+-+2%2Asqrt%2855%29%29%2F-10


Now break up the fraction


u=-10%2F-10%2B2%2Asqrt%2855%29%2F-10 or u=-10%2F-10-2%2Asqrt%2855%29%2F-10


Simplify


u=1-sqrt%2855%29%2F5 or u=1%2Bsqrt%2855%29%2F5


So these expressions approximate to

u=-0.483239697419133 or u=2.48323969741913


So our solutions are:
u=-0.483239697419133 or u=2.48323969741913

Notice when we graph -5%2Ax%5E2%2B10%2Ax%2B6 (just replace u with x), we get:



when we use the root finder feature on a calculator, we find that x=-0.483239697419133 and x=2.48323969741913.So this verifies our answer