Fun math problem if you are bored?

Favorite Answer

1. Solve the following equation √(x + 4) – √(x – 1) = 5 though the standard method.
√(x + 4) – √(x – 1) = 5 leads to no solution. Let's do
√(x + 4) + √(x – 1) = 5
√(x + 4) = 5 - √(x – 1)
x + 4 = 25 + x - 1 - 10√(x – 1)
10√(x – 1) = 20
x = 5
= = = = = = = =
2. Find another simpler and faster way to mathematically solve
√(x + 4) + √(x – 1) = 5
with no need to check for extraneous results.
Note (√(x + 4) + √(x – 1)) (√(x + 4) - √(x – 1)) = (x+4) - (x-1) = 5

√(x + 4) + √(x – 1) = (√(x + 4) + √(x – 1)) (√(x + 4) - √(x – 1))
√(x + 4) - √(x – 1) = 1
√(x + 4) + √(x – 1) = 5
√(x + 4) = 3 and √(x – 1) = 2
which both lead to x = 5
= = = = = = = =
3. By extending this second method, solve √(x + 4) + √(x – 1) = 10
√(x + 4) + √(x – 1) = 2*5 = 2 (√(x + 4) + √(x – 1)) (√(x + 4) - √(x – 1))
1 = 2 (√(x + 4) - √(x – 1))
√(x + 4) - √(x – 1) = 1/2
√(x + 4) + √(x – 1) = 10
√(x + 4) = 21/4 and √(x – 1) = 19/4
which both lead to x = 377/16
= = = = = = = =
4. Generalizing, give the full general solution to
√(x + A) + √(x – B) = C
where A, B and C are all real values.
(√(x + A) – √(x – B)) (√(x + A) + √(x – B)) = (x+A) - (x-B) = A+B
√(x + A) – √(x – B) = (C/(A+B)) (√(x + A) – √(x – B)) (√(x + A) + √(x – B))
(C/(A+B)) (√(x + A) – √(x – B)) = 1
√(x + A) – √(x – B) = (A+B)/C
√(x + A) + √(x – B) = C
√(x + A) = (A+B+C^2)/(2C)
x = (A+B+C^2)^2 / (4C^2) - A...Show more