Find two consecutive positive integers, the sum of whose squares is 365
Answers
Answer: 13, 14
Step-by-step explanation:
After the following steps, the answer is
Neglecting the negative value
x=13
The two consecutive no is x=13, x+1 =13+1 =14
sᴏʟᴜᴛɪᴏɴ:
ʟᴇᴛ ᴛʜᴇ ᴛᴡᴏ ᴄᴏɴsᴇᴄᴜᴛɪᴠᴇ ɴᴜᴍʙᴇʀs ʙᴇ x ᴀɴᴅ x+1.
ᴛʜᴇʀᴇғᴏʀᴇ ,
x² + (x+1)² = 365 x² + x² +1² + 2*x*1 = 365 (ʙᴇᴄᴀᴜsᴇ (ᴀ+ʙ)² = ᴀ² + ʙ²+ 2ᴀʙ)
ᴏʀ 2x² + 1 + 2x = 365
2x² + 2x = 365 - 1
2x² + 2x = 364
2(x² + x) = 364
ᴏʀ x² + x = 364/2
ᴏʀ x² + x = 182
ᴏʀ x² + x - 182 =0
ɴᴏᴡ sᴏʟᴠᴇ ᴛʜᴇ ϙᴜᴀᴅʀᴀᴛɪᴄ ᴇϙᴜᴀᴛɪᴏɴ ,
x² + 14x - 13x - 182 = 0
ɴᴏᴛᴇ : - 13 *14 = 182 , ᴛʜɪs ɪs ʙᴇᴄᴀᴜsᴇ ɪ ᴡʀɪᴛᴇ 14x - 13x ɪɴsᴛᴇᴀᴅ ᴏғ x , sᴏ ᴀs ᴛᴏ sᴏʟᴠᴇ ᴛʜᴇ ϙᴜᴀᴅʀᴀᴛɪᴄ ᴇϙᴜᴀᴛɪᴏɴ .
x (x+14) - 13 (x +14 ) = 0
( x- 13 )(x+14)=0
ᴛʜᴇʀᴇғᴏʀᴇ , ᴇɪᴛʜᴇʀ x - 13 = 0 ᴏʀ x+14 =0
sɪɴᴄᴇ ᴛʜᴇ ᴄᴏɴsᴇᴄᴜᴛɪᴠᴇ ɪɴᴛᴇɢᴇʀs ᴀʀᴇ ᴘᴏsɪᴛɪᴠᴇ ,
ᴛʜᴇʀᴇғᴏʀᴇ , x-13 = 0
⇒ x =13
ʜᴇɴᴄᴇ ᴏɴᴇ ᴏғ ᴛʜᴇ ᴘᴏsɪᴛɪᴠᴇ ɪɴᴛᴇɢᴇʀs = 13 ,
ᴛʜᴇʀᴇғᴏʀᴇ ᴏᴛʜᴇʀ ᴘᴏsɪᴛɪᴠᴇ ɪɴᴛᴇɢᴇʀ = x+1 = 13+1 = 14
sᴏ ᴛʜᴇ ᴛᴡᴏ ᴄᴏɴsᴇᴄᴜᴛɪᴠᴇ ᴘᴏsɪᴛɪᴠᴇ ɪɴᴛᴇɢᴇʀs ᴀʀᴇ 13 ᴀɴᴅ 14 .