Solving Equations with Rational Expressions?

Question

I really just don't get them. They hardly explained anything in the lesson. For instance:

a+1/2 + 3/2 = 1/a
They use the quadratic formula to solve it.
But in:
x/5 - 4/5 = 1/x
They dont.

I guess I'm just generally confused about it. If any of you can explain the steps and conditions in more detail Id appreciate it :)

Answer 1

a + 1/2 + 3/2 = 1/a
a + 2 = 1/a ................multiply by a
a² + 2a = 1
a² + 2a + 1 = 1 + 1 ......completing square
(a+1)² = 2
a + 1 = ±√2
a = -1 + √2 = 0.414 or
a = -1 - √2 = -2.414

x/5 - 4/5 = 1/x ........ mult by 5x
x² - 4x = 5
x² - 4x - 5 = 0
(x-5)(x+1) = 0
x = 5 or x = -1

I used completing the square on the first because it was easy; could just as well have used quad formula. But the second one factored. Where you can, factor, otherwise use completing the square or quad formula.

Answer 2

a+1/2+3/2=1/a is a quadratic equation, believe it or not.
This equation simplifies a bit: a+4/2=1/a
This becomes a+2=1/a
If we now clear out the a in the denominator , we can do it by multiplying both sides of the equation by a
a(a+2)=a(1/a)
a^2+2a=1
a^2+2a-1=0
This equation has the tell-tale sign of a quadratic: it contains a variable(in this case a) to the second power. Any time you see a ^2 term AND NOTHING HIGHER, you've got a quadratic
They used the quadratic formula to solve because
eye-balling a^2+2a-1 doesn't yield any obvious
factors. Had it been +1 instead of -1, the factors jump out as (a-1)(a-1)

x/5-4/5=1/x is also a quadratic
Multiply both sides by 5, x-4=5/x
Multiply again, both sides by x, x^2-4x=5
From this, x^2-4x-5=0, and this can be factored by
eye-balling.
(x-5)(x+1)=0
If you wanted to the quadratic formula could be used. It ALWAYS works. If, however, you can factor by eye-balling, you save time.
I see your confusion as perhaps one of when do you eye-ball, and when do you do the quadratic dance. The answer is try the eye-ball first. If you get nowhere, swing over to the quadratic formula. As I say, it ALWAYS works.
Good luck!

Answer 3

(a+1)/2 + 3/2 = 1/a
Multiply through by 2a

a(a+1) + 3a = 2
a^2 + a + 3a = 2
a^2 + 4a = 2

You can either complete the square or use the quadratic formula. Let's complete the square.

a^2 + 4a + 4 = 6
(a + 2)^2 = 6
(a + 2) = sqrt(6)
a = sqrt(6) - 2

x/5 - 4/5 = 1/x

Multiply through by 5x

x^2 - 4x = 5

Let's complete the square again.

x^2 - 4x + 4 = 5 + 4
(x - 2)^2 = 9
x - 2 = 3
x = 5