f the numerator and denominator of 13/17 are respectively increased and decreased by the same number , the resulting fraction is 1/2. Find the number.
Answers
Answer:
Solve for p:
p^2 (-p^2 + h^2) = a^2 h^2
Expand out terms of the left hand side:
-p^4 + h^2 p^2 = a^2 h^2
Subtract a^2 h^2 from both sides:
-a^2 h^2 + h^2 p^2 - p^4 = 0
Substitute x = p^2:
-a^2 h^2 + h^2 x - x^2 = 0
Multiply both sides by -1:
a^2 h^2 - h^2 x + x^2 = 0
Subtract a^2 h^2 from both sides:
x^2 - h^2 x = -a^2 h^2
Add h^4/4 to both sides:
x^2 - h^2 x + h^4/4 = h^4/4 - a^2 h^2
Write the both sides as a square:
(x - h^2/2)^2 = -1/4 h^2 (-h + 2 a) (h + 2 a)
Take the square root of both sides:
x - h^2/2 = 1/2 i sqrt(h^2 (-h + 2 a) (h + 2 a)) or x - h^2/2 = -1/2 i sqrt(h^2 (-h + 2 a) (h + 2 a))
Add h^2/2 to both sides:
x = h^2/2 + 1/2 i sqrt(h^2 (-h + 2 a) (h + 2 a)) or x - h^2/2 = -1/2 i sqrt(h^2 (-h + 2 a) (h + 2 a))
Substitute back for x = p^2:
p^2 = h^2/2 + 1/2 i sqrt(h^2 (-h + 2 a) (h + 2 a)) or x - h^2/2 = -1/2 i sqrt(h^2 (-h + 2 a) (h + 2 a))
Take the square root of both sides:
p = sqrt(h^2/2 + 1/2 i sqrt(h^2 (-h + 2 a) (h + 2 a))) or p = -sqrt(h^2/2 + 1/2 i sqrt(h^2 (-h + 2 a) (h + 2 a))) or x - h^2/2 = -1/2 i sqrt(h^2 (-h + 2 a) (h + 2 a))
h^2 (-h + 2 a) (h + 2 a) = -h^4 + 4 a^2 h^2:
p = sqrt(h^2/2 + 1/2 i sqrt(-h^4 + 4 a^2 h^2)) or p = -sqrt(h^2/2 + 1/2 i sqrt(h^2 (-h + 2 a) (h + 2 a))) or x - h^2/2 = -1/2 i sqrt(h^2 (-h + 2 a) (h + 2 a))
h^2 (-h + 2 a) (h + 2 a) = -h^4 + 4 a^2 h^2:
p = sqrt(h^2/2 + 1/2 i sqrt(-h^4 + 4 a^2 h^2)) or p = -sqrt(h^2/2 + 1/2 i sqrt(-h^4 + 4 a^2 h^2)) or x - h^2/2 = -1/2 i sqrt(h^2 (-h + 2 a) (h + 2 a))
Add h^2/2 to both sides:
p = sqrt(h^2/2 + 1/2 i sqrt(-h^4 + 4 a^2 h^2)) or p = -sqrt(h^2/2 + 1/2 i sqrt(-h^4 + 4 a^2 h^2)) or x = h^2/2 - 1/2 i sqrt(h^2 (-h + 2 a) (h + 2 a))
Substitute back for x = p^2:
p = sqrt(h^2/2 + 1/2 i sqrt(-h^4 + 4 a^2 h^2)) or p = -sqrt(h^2/2 + 1/2 i sqrt(-h^4 + 4 a^2 h^2)) or p^2 = h^2/2 - 1/2 i sqrt(h^2 (-h + 2 a) (h + 2 a))
Take the square root of both sides:
p = sqrt(h^2/2 + 1/2 i sqrt(-h^4 + 4 a^2 h^2)) or p = -sqrt(h^2/2 + 1/2 i sqrt(-h^4 + 4 a^2 h^2)) or p = sqrt(h^2/2 - 1/2 i sqrt(h^2 (-h + 2 a) (h + 2 a))) or p = -sqrt(h^2/2 - 1/2 i sqrt(h^2 (-h + 2 a) (h + 2 a)))
h^2 (-h + 2 a) (h + 2 a) = -h^4 + 4 a^2 h^2:
p = sqrt(h^2/2 + 1/2 i sqrt(-h^4 + 4 a^2 h^2)) or p = -sqrt(h^2/2 + 1/2 i sqrt(-h^4 + 4 a^2 h^2)) or p = sqrt(h^2/2 - 1/2 i sqrt(-h^4 + 4 a^2 h^2)) or p = -sqrt(h^2/2 - 1/2 i sqrt(h^2 (-h + 2 a) (h + 2 a)))
h^2 (-h + 2 a) (h + 2 a) = -h^4 + 4 a^2 h^2:
Answer: |
| p = sqrt(h^2/2 + 1/2 i sqrt(-h^4 + 4 a^2 h^2)) or p = -sqrt(h^2/2 + 1/2 i sqrt(-h^4 + 4 a^2 h^2)) or p = sqrt(h^2/2 - 1/2 i sqrt(-h^4 + 4 a^2 h^2)) or p = -sqrt(h^2/2 - 1/2 i sqrt(-h^4 + 4 a^2 h^2))