Question

The number of integral values of a for which the equation x2 + (2a - 4)x + (2a - 12)=0

Ohas rational roots, is

Zero
0 1
02
infinitely many​

Answer 1

Given :

Equation - x^2 + (2a - 4)x + (2a - 12)=0

To find :

The number of integral values of a for which the equation has rational roots

Solution :

• Sum of roots = α + β

= 2a - 4

• Products of roots = α × β

= 2a - 12

• (α - β)^2 = (α + β)^2 - 4αβ

(α - β)^2 = (2a - 4)^2 - 4×(2a - 12)

= 4×[ (a - 2)^2 - 2(a- 3) ]

= 4×[ a^2 - 6a + 10 ]

= 4×[ a^2 - 2×3a + 9 + 1 ]

= 4×[ (a - 3)^2 + 1 ]

= 4×(a - 3)^2 + 4

• We can see that for (α - β)^2 to be rational number, a can take infinitely many solutions.

• Hence, number of integral values of a are infinitely many