Question

The equations x2− 4x + k = 0 and x2+ kx − 4 = 0, where k is a real number, have exactly one common root. What is the value Ofp k?

Answer 1

x² − 4x + k = 0
Roots: a and c
ac = k ----> c = k/a
a + c = 4
a + k/a = 4
k = a(4−a)

x² + kx − 4 = 0
Roots: a and b
ab = −4 ----> b = −4/a
a + b = −k
a − 4/a = −k
k = 4/a − a = (4−a²)/a

a(4−a) = (4−a²)/a
a²(4−a) = (4−a²)
4a² − a³ = 4 − a²
a³ − 5a² + 4 = 0
(a − 1) (a² − 4a − 4) = 0
a = 1, 2 ± 2√2

a = 1
k = a(4−a) = 3
b = −4/a = −4
c = k/a = 3

a = 2+2√2
k = a(4−a) = −4
b = −4/a
c = k/a = −4/a = b

a = 2−2√2
k = a(4−a) = −4
b = −4/a
c = k/a = −4/a = b

k = 3

x² − 4x + 3 = 0 ----> (x−1)(x−3) = 0 ----> x = 1 or 3
x² + 3x − 4 = 0 ----> (x−1)(x+4) = 0 ----> x = 1 or −4