SOLUTION: Find the solutions to the equation x-5/x= 4.

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Question 102555This question is from textbook Algebra and Trigonometry
: Find the solutions to the equation x-5/x= 4. This question is from textbook Algebra and Trigonometry

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
x-5%2Fx=+4 Start with the given equation


x%5E2-5=+4x Multiply both sides by the LCD x


x%5E2-4x-5=0+ Subtract 4x from both sides


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 x%5E2-4%2Ax-5=0 ( notice a=1, b=-4, and c=-5)




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



x+=+%284+%2B-+sqrt%28+%28-4%29%5E2-4%2A1%2A-5+%29%29%2F%282%2A1%29 Negate -4 to get 4



x+=+%284+%2B-+sqrt%28+16-4%2A1%2A-5+%29%29%2F%282%2A1%29 Square -4 to get 16 (note: remember when you square -4, you must square the negative as well. This is because %28-4%29%5E2=-4%2A-4=16.)



x+=+%284+%2B-+sqrt%28+16%2B20+%29%29%2F%282%2A1%29 Multiply -4%2A-5%2A1 to get 20



x+=+%284+%2B-+sqrt%28+36+%29%29%2F%282%2A1%29 Combine like terms in the radicand (everything under the square root)



x+=+%284+%2B-+6%29%2F%282%2A1%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)



x+=+%284+%2B-+6%29%2F2 Multiply 2 and 1 to get 2

So now the expression breaks down into two parts

x+=+%284+%2B+6%29%2F2 or x+=+%284+-+6%29%2F2

Lets look at the first part:

x=%284+%2B+6%29%2F2

x=10%2F2 Add the terms in the numerator
x=5 Divide

So one answer is
x=5



Now lets look at the second part:

x=%284+-+6%29%2F2

x=-2%2F2 Subtract the terms in the numerator
x=-1 Divide

So another answer is
x=-1

So our solutions are:
x=5 or x=-1

Notice when we graph x%5E2-4%2Ax-5, we get:

+graph%28+500%2C+500%2C+-11%2C+15%2C+-11%2C+15%2C1%2Ax%5E2%2B-4%2Ax%2B-5%29+

and we can see that the roots are x=5 and x=-1. This verifies our answer