(i) $2x^{2} + kx$ + 3 = 0
(ii) kx (x – 2) + 6 = 0
(i) $2x^{2} + kx + 3$ = 0
Comparing the given equation with $ax^{2} + bx + c$ = 0, we get
a = 2, b = k and c = 3
As we know, discriminant =$ b^{2} – 4ac$
= $(k)^{2} $– 4(2) (3)
=$ k^{2}$ – 24
For equal roots, we know,
Discriminant = 0
$k^{2}$ – 24 = 0
$k^{2}$ = 24
k = ±$\sqrt{24}$ = ±$2\sqrt{6}$
(ii) kx(x – 2) + 6 = 0
or $kx^{2}$ – 2kx + 6 = 0
Comparing the given equation with $ax^{2}+ bx + c$ = 0, we get
a = k, b = – 2k and c = 6
We know, Discriminant = $b^{2} – 4ac$
=$ ( – 2k)^{2} $– 4 (k) (6)
= $4k^{2}$ – 24k
For equal roots, we know,
$b^{2} – 4ac$ = 0
$4k^{2} – 24k$ = 0
4k (k – 6) = 0
Either 4k = 0 or k = 6 = 0
k = 0 or k = 6
However, if k = 0, then the equation will not have the terms ‘$x^{2}$‘and ‘x‘.
Therefore, if this equation has two equal roots, k should be 6 only.