Question

If 2 is the root of the quadratic equation 3x2-2px-8=0 and the quadratic equation 4x2-2px+k=0 has equal roots . Find k

Answer 1

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  

Answer 2

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 .


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