find the value of k for which the quadratic equation (3k+1)x2 + 2 (k+1) x+1=0 has equal roots also find the roots
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ʜᴇʀᴇ ɪs ʏᴏᴜʀ ᴀɴsᴡᴇʀ :-
( 3ᴋ + 1 )x² + 2 ( ᴋ + 1 ) + 1 = 0
ᴄᴏᴍᴘᴀʀɪɴɢ ᴛʜᴇ ᴇǫᴜᴀᴛɪᴏɴ ᴡɪᴛʜ sᴛᴀɴᴅᴀʀᴅ ғᴏʀᴍ
ʜᴇʀᴇ
ᴀ = 3ᴋ + 1
ʙ = 2( ᴋ + 1 )
ᴄ = 1
sɪɴᴄᴇ ᴛʜᴇ ɢɪᴠᴇɴ ǫᴜᴀᴅʀᴀᴛɪᴄ ᴇǫᴜᴀᴛɪᴏɴ ʜᴀs ᴇǫᴜᴀʟ ʀᴏᴏᴛs ᴛʜᴇ ᴅɪsᴄʀɪᴍɪɴᴀɴᴛ = 0
ʙ² - 4ᴀᴄ = 0
( 3ᴋ + 1 )² - 4 ×2( ᴋ + 1 )( 1 ) = 0
=> 9ᴋ² + 6ᴋ + 1 - 8ᴋ - 8 = 0
=> 9ᴋ² - 2ᴋ - 7 = 0
=> 9ᴋ² - 9ᴋ + 7ᴋ - 7 = 0
=> 9ᴋ ( ᴋ - 1 ) + 7 ( ᴋ - 1 ) = 0
=> ( ᴋ - 1 )( 9ᴋ + 7 ) = 0
=> ᴋ - 1 = 0. ᴏʀ. 9ᴋ + 7. = 0
=> ᴋ = 1 ᴏʀ. ᴋ = -7/9
( 3ᴋ + 1 )x² + 2 ( ᴋ + 1 ) + 1 = 0
ᴄᴏᴍᴘᴀʀɪɴɢ ᴛʜᴇ ᴇǫᴜᴀᴛɪᴏɴ ᴡɪᴛʜ sᴛᴀɴᴅᴀʀᴅ ғᴏʀᴍ
ʜᴇʀᴇ
ᴀ = 3ᴋ + 1
ʙ = 2( ᴋ + 1 )
ᴄ = 1
sɪɴᴄᴇ ᴛʜᴇ ɢɪᴠᴇɴ ǫᴜᴀᴅʀᴀᴛɪᴄ ᴇǫᴜᴀᴛɪᴏɴ ʜᴀs ᴇǫᴜᴀʟ ʀᴏᴏᴛs ᴛʜᴇ ᴅɪsᴄʀɪᴍɪɴᴀɴᴛ = 0
ʙ² - 4ᴀᴄ = 0
( 3ᴋ + 1 )² - 4 ×2( ᴋ + 1 )( 1 ) = 0
=> 9ᴋ² + 6ᴋ + 1 - 8ᴋ - 8 = 0
=> 9ᴋ² - 2ᴋ - 7 = 0
=> 9ᴋ² - 9ᴋ + 7ᴋ - 7 = 0
=> 9ᴋ ( ᴋ - 1 ) + 7 ( ᴋ - 1 ) = 0
=> ( ᴋ - 1 )( 9ᴋ + 7 ) = 0
=> ᴋ - 1 = 0. ᴏʀ. 9ᴋ + 7. = 0
=> ᴋ = 1 ᴏʀ. ᴋ = -7/9
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