114k views
19 votes
[tex]\sqrt{9-4\sqrt{5} }\:+\sqrt{5}

The answer is 2. Solve if you can. It is 100% correct.

2 Answers

6 votes


\\ \rm\Rrightarrow \sqrt{9-4√(5)}+√(5)


\\ \rm\Rrightarrow √(9-8.8)+2.2


\\ \rm\Rrightarrow √(0.2)+2.2


\\ \rm\Rrightarrow 0.3+2.2


\\ \rm\Rrightarrow 2.5


\\ \rm\Rrightarrow 2(approx)

User Cory Robinson
by
8.1k points
7 votes

Answer:

Algebraic "proof" that the solution is 2:


\sqrt{9-4√(5) }\:+√(5)


=\sqrt{4-4√(5)+5 }\:+√(5)


=\sqrt{2^2-4√(5)+(√(5))^2 }\:+√(5)


=\sqrt{(2-√(5))^2 }\:+√(5)


=(2-√(5))+√(5)


=2

However, this is not the correct mathematical solution to the problem.

The order of operations for
\sqrt{(2-√(5))^2 } dictate that the operation inside the parentheses must be carried out first.

As
2-√(5) < 0, then
(2-√(5))^2 will always be positive.

If
(2-√(5))^2 is always positive, then
\sqrt{(2-√(5))^2 } will always be positive.

As √5 > 2 and
\sqrt{(2-√(5))^2 } > 0 then
\sqrt{(2-√(5))^2 }\:+√(5) > 2

So mathematically, the actual solution to the expression is 2.47 (nearest hundredth) not 2.

User Honzajde
by
7.8k points

Related questions

asked Dec 18, 2024 7.9k views
TomerBu asked Dec 18, 2024
by TomerBu
7.7k points
1 answer
3 votes
7.9k views
asked Jan 13, 2024 11.7k views
Venkat Reddy asked Jan 13, 2024
by Venkat Reddy
8.4k points
2 answers
2 votes
11.7k views
asked Aug 20, 2024 142k views
Sshturma asked Aug 20, 2024
by Sshturma
8.0k points
1 answer
2 votes
142k views