Question

If 2 is a root of the quadratic equation 3x2 + px − 8 = 0 and the quadratic equation 4x2 − 2px + k = 0 has equal roots, find the value of k.

Answer 1

the value of k is -24

Answer 2

2 ɪs ᴛʜᴇ ʀᴏᴏᴛ ᴏғ ǫᴜᴀᴅʀᴀᴛɪᴄ ᴇǫᴜᴀᴛɪᴏɴ 3x² + ᴘx - 8 = 0………….(1)

& 4x² - 2ᴘx + ᴋ = 0 ʜᴀs ᴇǫᴜᴀʟ ʀᴏᴏᴛs  ………….(2)

ᴏɴ ᴘᴜᴛᴛɪɴɢ ᴛʜᴇ ᴠᴀʟᴜᴇ ᴏғ ɢɪᴠᴇɴ ʀᴏᴏᴛ ɪ.ᴇ x = 2 ɪɴ ᴇǫ 1 .

3x² + ᴘx - 8 = 0

3(2)² + ᴘ(2) − 8 = 0

3 × 4 + 2ᴘ - 8 = 0

12 + 2ᴘ − 8 = 0

4 + 2ᴘ = 0

2ᴘ = - 4

ᴘ = - 4/ 2  = -2

ᴘ = - 2

ʜᴇɴᴄᴇ ᴛʜᴇ ᴠᴀʟᴜᴇ ᴏғ ᴘ ɪs - 2.
ᴏɴ ᴘᴜᴛᴛɪɴɢ ᴛʜᴇ ᴠᴀʟᴜᴇ ᴏғ ᴘ = - 2 ɪɴ ᴇǫ 2,  

4x² - 2ᴘx + ᴋ = 0

4x² - 2(- 2)x + ᴋ = 0

4x² + 4x + ᴋ = 0

ᴏɴ ᴄᴏᴍᴘᴀʀɪɴɢ ᴛʜᴇ ɢɪᴠᴇɴ ᴇǫᴜᴀᴛɪᴏɴ ᴡɪᴛʜ ᴀx² + ʙx + ᴄ = 0  

ʜᴇʀᴇ, ᴀ = 4, ʙ =  4 , ᴀɴᴅ ᴄ = ᴋ

ᴅ(ᴅɪsᴄʀɪᴍɪɴᴀɴᴛ) = ʙ² – 4ᴀᴄ

ɢɪᴠᴇɴ : ǫᴜᴀᴅʀᴀᴛɪᴄ ᴇǫᴜᴀᴛɪᴏɴ ʜᴀs ᴇǫᴜᴀʟ ʀᴏᴏᴛs ɪ.ᴇ ᴅ = 0  

ʙ² – 4ᴀᴄ = 0

4² – 4(4)(ᴋ) = 0

16 – 16ᴋ = 0

16 = 16ᴋ

ᴋ = 16/16  = 1

ᴋ = 1

ʜᴇɴᴄᴇ, ᴛʜᴇ ᴠᴀʟᴜᴇ ᴏғ ᴋ  ɪs 1 .
★★ ɴᴀᴛᴜʀᴇ ᴏғ ᴛʜᴇ ʀᴏᴏᴛs

ɪғ ᴅ = 0 ʀᴏᴏᴛs ᴀʀᴇ ʀᴇᴀʟ ᴀɴᴅ ᴇǫᴜᴀʟ  

ɪғ ᴅ > 0 ʀᴏᴏᴛs ᴀʀᴇ ʀᴇᴀʟ ᴀɴᴅ ᴅɪsᴛɪɴᴄᴛ

ɪғ ᴅ < 0  ɴᴏ ʀᴇᴀʟ ʀᴏᴏᴛs