Answer:
-7 and -5
Explanation:
let a be an odd integer.
Then ‘a’ can be written as :
a = 2p + 1 ; where p is an integer.
The next odd number to ‘a’ is :
a + 2 = 2p + 1 = 2p + 3
The statement “the square of the first, added to 3 times the second,
is 34”
means
a² + 3(a + 2) = 34
………………………………………………
Solving the equation :
a² + 3(a + 2) = 34
⇔ (2p + 1)² + 3(2p + 3) = 34
⇔ 4p² + 4p + 1 + 6p + 9 = 34
⇔ 4p² + 10p + 10 = 34
⇔ 4p² + 10p - 24 = 0
⇔ 2p² + 5p - 12 = 0
⇔ 2p² + 5p +3p - 3p - 12 = 0
⇔ 2p² + 8p - 3(p + 4) = 0
⇔ 2p(p + 4) - 3(p + 4) = 0
⇔ (p + 4) (2p - 3) = 0
⇔ p + 4 = 0 or 2p - 3 = 0
⇔ p = -4 or p = 3/2
(3/2 to be rejected because not an integer)
Then
p = -4
Calculating a and a+2 :
a = 2p + 1 = 2(-4) + 1 = -7
a + 2 = 2p + 3 = 2(-4) + 3 = -5
Verification :
(-7)² + 3(-5) = 49 - 15 = 34