a. p = 4
b. m = 16
c. k = 3  atau  k = –3.
d. m = 8  atau  m = –8.
e. k = 4/3 atau k = –4/3.
f. p = 16 atau p = 4.
g. k = 1  atau  k = –1/3.
h. m = –3/2  atau  m = 3.
 

Pembahasan

Akar kembar/sama dimiliki persamaan kuadrat ax² + bx + c = 0 jika b² = 4ac terpenuhi.

a. px² – 12x + 9 = 0
144 = 36p
p = 144/36 = 4
_____________

b. x² – 8x + m = 0
64 = 4m
m = 64/4 = 16
_____________

c. x² + 2kx + 9 = 0
4k² = 36
k = ±√(36/4) = ±3
k = 3  atau  k = –3.
_____________

d. x² + mx + 16 = 0
m² = 64
m = ±8
m = 8  atau  m = –8.
_____________

e. x² + (kx – 5)² = 9
Jabarkan dulu.
x² + k²x² – 10kx + 25 – 9 = 0
(1 + k²) – 10kx + 16 = 0
Lalu, b² = 4ac.
100k² = 64 + 64k²
36k² = 64
k = √(64/36) = 8/6 = ±4/3
k = 4/3  atau  k = –4/3.
_____________

f. px² + (p + 8)x = –9
(p + 8)² = 36p
p² + 16p + 64 – 36p = 0
p² – 20p + 64 = 0
(p – 16)(p – 4) = 0
p = 16  atau  p = 4.
_____________

g. k²x² + 2(k + 1)x + 4 = 0
4(k + 1)² = 16k²
(k + 1)² = 4k²
(k + 1)² = (±2k)²
k + 1 = 2k atau k + 1 = –2k
k = 1  atau  k = –1/3.
_____________

h. (m + 1)x² – 2(m + 3)x + 3m = 0​
4(m + 3)² = 12m(m + 1)
(m + 3)² = 3m(m + 1)
m² + 6m + 9 = 3m² + 3m
2m² – 3m – 9 = 0
(2m + 3)(m – 3) = 0
m = –3/2  atau  m = 3.