Question

1. Find the values of $k$ for which the following equations have real and equal roots:
$\begin{array}{c}\text{(i)}{x}^2-2(k+1)x+k^2=0\\ \text{(ii)}{k}^2{x}^2-2(2k-1)x+4=0\\ \text{(iii)}(k+1)x^2-2(k-1)x+1=0\\ \text{(iv)}{x}^2+k(2x+k-1)+2=0\end{array}$
2. Find the values of $k$ for which the following equations have real roots:
$\begin{array}{c}\text{(i)}2x^2+kx+3=0\\ \text{(ii)}{x}^2-4kx+k=0\\ \text{(iii)}kx(x-3)+9=0\end{array}$
$\begin{array}{c}\text{(iv)}kx(x-2)+6=0\\ \text{(v)}kx(x-2\sqrt{5})+10=0\\ \text{(vi)}4x^2+kx+3=0\end{array}$

Answer 1

$i)x^2-2(k+1)x+k^2=0$

$D=0$

$\Rightarrow 4(k+1{)}^2-4k^2=0$

$\Rightarrow 4(k+1-k)(k+1+k)=0$

$\Rightarrow 2k+1=0\Rightarrow k=-1/2$

$ii)k^2{x}^2-2(k-1)x+4=0$

$D=0$

$\Rightarrow 4(2k-1{)}^2-4k^2.4=0$

$\Rightarrow 4(2k-1-2k)(2k-1+2k)=0$

$\Rightarrow -4(4k-1=0\Rightarrow k=1/4$

$iii\left)\right(k+1)x^2-2(k-1)x+1=0$

$D=0$

$\Rightarrow 4(k-1{)}^2-4(k+1{)}^2=0$

$\Rightarrow 4(k-1-k-1)(k-1+k+1)=0$

$\Rightarrow k=0$

$iv)x^2+k(2x+k-1)+2=0$

$\Rightarrow x^2+2kx+k^2-k+2=0$

$D=0$

$\Rightarrow 4k^2-4(k^2-k+2)=0$

$\Rightarrow 4(k^2-k^2+k-2)=0$

$\Rightarrow k=2$

2. $i)2x^2+kx+3=0$

$D\ge =0$

$\Rightarrow k^2-\mathrm{4.2.3}\ge 0$

$\Rightarrow k^2\ge 24$

$\Rightarrow k\le -2\sqrt{6}|k\ge 2\sqrt{6}$

$ii)x^2-4kx+k=0$

$\Rightarrow 16k^2-4k\ge 0$

$\Rightarrow 4k(4k-1)\ge 0$

$\Rightarrow k(k-1/4)\ge 0$

Refer image.

$\Rightarrow k\le 0$ or $k\ge 1/4$

$iii\left)kx\right(x-3)+9=0$

$k{x}^2-3kx+9=0$

$D\ge 0$

$9k^2-4.k.9\ge 0$

$9k(k-4)\ge 0$

$\Rightarrow k\le 0$ or $k\ge 4$

$iv\left)kx\right(x-2)+6=0$

$k{x}^2-2kx+6=0$

$D\ge 0$

$\Rightarrow 4k^2-4.k.6\ge 0$

$\Rightarrow 4k(k-6)\ge 0$

$\Rightarrow k\le 0$ or $k\ge 6$

$v\left)kx\right(x-2\sqrt{5})+10=0$

$\Rightarrow k{x}^2-2\sqrt{5}kx+10=0$

$D\ge 0$

$\Rightarrow \mathrm{4.5.}{k}^2-4.k.10\ge 0$

$\Rightarrow 20k(k-2)\ge 0$

$\Rightarrow k\le 0$ or $k\ge 2$

$vi)4x^2+kx+3=0$

$D\ge 0$

$\Rightarrow k^2-\mathrm{4.4.3}\ge 0$

$\Rightarrow (k+4\sqrt{3})(k-4\sqrt{3})\ge 0$

$k\le -4\sqrt{3}$ or $k\ge 4\sqrt{3}$