Quadratics formula?

Question

Ok once again I find myself dazed and confused.....I know I have to use the quadratic formula to answer this....but...I am so lost...

6/x-4 = x-4/x

Restricted value....I think is X cannot be 4 or -1

11 pages later I get 11 different answers (sigh)

Answer 1

Right again on restrictions. Here you cross-multiply:
(x-4)(x-4) =6x

x^2 - 8x + 16 = 6x

Subtract 6x to make it = 0

x^2 - 14x + 16 = 0

You can't do FOIL so use the quadratic formula with
a=1, b= -14 and c= 16

14 +- sqrt(196-4x1x16) all over 2x1

14 +- sqrt 132 all over 2

sqrt 132 simplifies to 2 sqrt 33

14 +- 2 sqrt 33 all over 2

That's 7 +- sqrt 33

Check the work

Answer 2

Actually does your original problem look like this?

6.............4
- - 4 = x - -
x..............x

If so, then the answer below is correct.

The only value x can't be is zero, because that would result in zeros in the denominator in the original equation, and that's forbidden.

Multiply both sides by x:

6 - 4x = x^2 - 4

Get everything on one side:

0 = x^2 + 4x -10

Use quadratic formula:

x = [-4 +/- sqrt(16 - 4(1)(-10))]/2(1)
x = [-4 +/- sqrt(56)]/2
x = [-4 +/- sqrt((4)(14)]/2
x = [-4 +/- 2sqrt(14)]/2
x = -2 +/- sqrt(14)

Ugly, but that's the answer, if I didn't mess up.

If instead, the original problem looked like:

6..........x - 4
------ = ------
x - 4......x

(and if it did, you should have written it as 6/(x-4) = (x-4)/x)

Then x can't be equal to 4 or 0, and you do cross-multiply

(x-4)(x-4) = 6x
x^2 - 8x + 16 = 6x
x^2 - 14x + 16 = 0

then use the quadratic on that...

Answer 3

x = 0 is the undefined value. With that out of the way...

Multiply everything by x to get rid of the demoninator (I call them demoninators because they are evil) That leaves you with:

6 - 4x = x^2 - 4
0 = x^2 + 4x - 10
x = (-4 +- sqrt(16 + 40))/2 = -2 +- sqrt(14)

Answer 4

6/(x-4) = (x-4)/x
Simplest way is to cross multiply, getting:
(x-4)(x-4) = 6x
x^2 -16X +16 =6x
x^2 - 22x +16 =0
Now use quadratic formula:
x= [22 + or - sqrt( 22^2-4*1*16)]/2
x= [22+ 2sqrt(105)]/2 = 11+ sqrt(105), and
x = 11-sqrt(105)

Answer 5

You multiplies both sides with x and at the same times notes that x can't be 0 cause x is in the denominator in the original equation and division by zero is undefined.

Now you have yourself a second order equation.

Answer 6

x cannot equal 0 since you cannot divide by 0.

6/x - 4 = x - 4/x

Multiply both sides by x.

6 - 4x = x^2 - 4
0 = x^2 + 4x - 10

x = (-4 +/- sqrt(16 + 40))/2
= -2 +/- sqrt(14)

Answer 7

(x-4)^2=6x

x^2 + 16 -8x = 6x

X^2 +16 -14x = 0

x = 7 (+or -) sqrt(33)