Question

If x2 - ax - 6 = 0 and x2 + ax - 2 = 0 have a common root then a = ?
a) +1
b) 2
3)-3
d) o
​

Answer 1

Answer:

option 3) –3

Step-by-step explanation:

Given :

x² - ax - 6 = 0 and x² + ax - 2 = 0 have a common root.

To find :

the value of 'a'

Solution :

To find the value of a, we must know the relationship between zeroes and coefficients.

sum of zeroes = –(x coefficient) /x² coefficient

product of zeroes = constant term/x² coefficient

Let c, d are the zeroes of the polynomial x² – ax – 6 = 0

Sum of zeroes :

c + d = -(-a)/1 = a

d = a – c

Product of zeroes :

cd = –6

Put d = a – c,

(c)(a – c) = –6

ca – c² = –6

ca = c² – 6 --(1)

Let c, e are the zeroes of the polynomial x² + ax – 2 = 0

Sum of zeroes :

c + e = –a

e = –a – c

e = –(a + c)

Product of zeroes :

ce = –2

Put e = –(a + c)

(c) (–[a + c]) = –2

(c) (a + c) = 2

ac + c² = 2

Put ac = c² – 6, (eqn.[1])

c² – 6 + c² = 2

2c² = 2 + 6

2c² = 8

c² = 8/2

c² = 4

c = √4

c = ±2

Therefore, c = 2, –2

If c = 2,

ac + c² = 2

a(2) + 4 = 2

2a = 2 – 4

2a = –2

a = –2/2

a = –1

If c = –2,

ac + c² = –2

a(2) + 4 = –2

2a = –2 – 4

2a = –6

a = –6/2

a = –3

Therefore, a can be –1 or –3