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.
Answers
SOLUTION :
Given : 2 is the root of quadratic equation 3x² + px - 8 = 0………….(1)
& 4x² - 2px + k = 0 has equal roots ………….(2)
On putting the value of given root i.e x = 2 in eq 1 .
3x² + px - 8 = 0
3(2)² + p(2) − 8 = 0
3 × 4 + 2p - 8 = 0
12 + 2p − 8 = 0
4 + 2p = 0
2p = - 4
p = - 4/ 2 = -2
p = - 2
Hence the value of p is - 2.
On putting the value of p = - 2 in eq 2,
4x² - 2px + k = 0
4x² - 2(- 2)x + k = 0
4x² + 4x + k = 0
On comparing the given equation with ax² + bx + c = 0
Here, a = 4, b = 4 , and c = k
D(discriminant) = b² – 4ac
Given : Quadratic equation has equal roots i.e D = 0
b² – 4ac = 0
4² – 4(4)(k) = 0
16 – 16k = 0
16 = 16k
k = 16/16 = 1
k = 1
Hence, the value of k is 1 .
★★ NATURE OF THE ROOTS
If D = 0 roots are real and equal
If D > 0 roots are real and distinct
If D < 0 No real roots
HOPE THIS ANSWER WILL HELP YOU…
& 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