Question

The values of $k$ for which the roots are real and equal of the following equation

$x^2-2(5+2k)x+3(7+10k)=0$ are $k=2,\frac{1}{2}$

• A. True
• B. False

Answer 1

The correct option is A True

$x^2-2(5+2k)k+3(7+10k)=0$

$\Rightarrow$ Here, $a=1,b=-2(5+2k),c=3(7+10k)$

$\Rightarrow$ It is given that roots are real and equal.

$\therefore$ $b^2-4ac=0$

$\Rightarrow$ $[-2(5+2k){]}^2-4(1\left)\right[3(7+10k)]=0$

$\Rightarrow$ $4(25+20k+4k^2)-4(21+30k)=0$

$\Rightarrow$ $100+80k+16k^2-84-120k=0$

$\Rightarrow$ $16k^2-40k+16=0$

$\Rightarrow$ $2k^2-5k+2=0$

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

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

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

$\Rightarrow$ $k-2=0$ and $2k-1=0$

$\therefore$ $k=2$ and $k=\frac{1}{2}$