SOLUTION: Could you please help me with the following problem? I am supposed to solve this equation using the quadratic formula. (1/y)-(1/y+5)=(1/4) Thank you very much for any help you

Algebra ->  Quadratic Equations and Parabolas -> SOLUTION: Could you please help me with the following problem? I am supposed to solve this equation using the quadratic formula. (1/y)-(1/y+5)=(1/4) Thank you very much for any help you      Log On


   

Question 88959: Could you please help me with the following problem? I am supposed to solve this equation using the quadratic formula. (1/y)-(1/y+5)=(1/4)
Thank you very much for any help you can provide.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
1%2Fy-1%2F%28y%2B5%29=1%2F4

4y%28y%2B5%29%281%2Fy-1%2F%28y%2B5%29%29=4y%28y%2B5%29%281%2F4%29 Multiply both sides by the LCD 4y(y+5)

4%28y%2B5%29-4y=y%28y%2B5%29 Distribute

4y%2B20-4y=y%5E2%2B5y Distribute again

20=y%5E2%2B5y Combine like terms

0=y%5E2%2B5y-20 Subtract 20 from both sides



Now let's use the quadratic formula to solve for y:


Starting with the general quadratic

ay%5E2%2Bby%2Bc=0

the general solution using the quadratic equation is:

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

So lets solve y%5E2%2B5%2Ay-20=0 ( notice a=1, b=5, and c=-20)

y+=+%28-5+%2B-+sqrt%28+%285%29%5E2-4%2A1%2A-20+%29%29%2F%282%2A1%29 Plug in a=1, b=5, and c=-20



y+=+%28-5+%2B-+sqrt%28+25-4%2A1%2A-20+%29%29%2F%282%2A1%29 Square 5 to get 25



y+=+%28-5+%2B-+sqrt%28+25%2B80+%29%29%2F%282%2A1%29 Multiply -4%2A-20%2A1 to get 80



y+=+%28-5+%2B-+sqrt%28+105+%29%29%2F%282%2A1%29 Combine like terms in the radicand (everything under the square root)



y+=+%28-5+%2B-+sqrt%28105%29%29%2F%282%2A1%29 Simplify the square root



y+=+%28-5+%2B-+sqrt%28105%29%29%2F2 Multiply 2 and 1 to get 2

So now the expression breaks down into two parts

y+=+%28-5+%2B+sqrt%28105%29%29%2F2 or y+=+%28-5+-+sqrt%28105%29%29%2F2


which split up to


y=-5%2F2%2Bsqrt%28105%29%2F2 or y=-5%2F2-sqrt%28105%29%2F2




Which approximate to

y=2.6234753829798 or y=-7.6234753829798


So our solutions are:
y=2.6234753829798 or y=-7.6234753829798

Notice when we graph x%5E2%2B5%2Ax-20 (just replace y with x), we get:



when we use the root finder feature on a calculator, we find that x=2.6234753829798 and x=-7.6234753829798.So this verifies our answer