Algebra Problem:

Question

Simplify: (4-√20)/(2-2√5)

(A) (3-√5)/(4)

(B) 2

(C) (3+2√5)/(4)

(D) 1

(E) None of these

For this I got; (A) (3-√5)/(4). Is this the correct answer?

Answer 1

^{4 - √20} / _{2 - 2√5}

Let's rework that sqrt 20:

√20 ⇒ √(4 * 5) ⇒ 2√5

Now let's remove that square root in the denominator by using its conjugate:

^{(4 - 2√5)} / _{(2 - 2√5)} * ^{(2 + 2√5)} / _{(2 + 2√5)}

Now to FOIL the fraction.

Numerator:

(4 - 2√5) * (2 + 2√5)

8 + 8√5 - 4√5 - 20

4√5 - 12

denominator

(2 - 2√5) * (2 + 2√5)

4 + 4√5 - 4√5 - 4 * 5

4 - 20 = -16

So that gives us:

^{4√5 - 12} / _-16

Let's multiply by -1:

^{12 - 4√5} / ₁₆

We can factor out a 4:

^{4(3 - √5)} / _{4 * 4}

^{3 - √5} / ₄

So that does indeed seem to be correct. If we want to check, Let's compare the above with the simplified in a calculator:

^{4 - √20} / _{2 - 2√5} = 0.19098300562505257589770658281718

^{3 - √5} / ₄ = 0.19098300562505257589770658281718

So yes, choice A is the correct answer.

Answer 2

In this problem, we will make use of the identity (a - b)(a + b) = a² - b²

(4-√20) (4-√20) (2+2√5) 8 + 8√5 - 2√20 - 2√100 8 + 8√5 - 2√(4*5) - 2*10

------------ = ---------- --------------- = ------------------------------------- = --------------------------------------

(2-2√5) (2-2√5) (2+2√5) 4 - (2√5)^{2 4 - 4*5}

8 + 8√5 - 2*2√5 - 20 -12 + 4√5 3 - √5

------------------------------ = ----------------- = -------------

-16 -16 4

Answer: A