Find the Value of k for Which the Roots Are Real and Equal
Solution
Given: $$x^2-2(5+2k)x+3(7+10k)=0$$
Here, $$a=1,\quad b=-2(5+2k),\quad c=3(7+10k)$$
For real and equal roots, $$D=b^2-4ac=0$$
$$[-2(5+2k)]^2-4(1)\,[3(7+10k)]=0$$
$$4(5+2k)^2-12(7+10k)=0$$
$$(5+2k)^2-3(7+10k)=0$$
$$25+20k+4k^2-21-30k=0$$
$$4k^2-10k+4=0$$
$$2k^2-5k+2=0$$
$$(2k-1)(k-2)=0$$
$$k=\frac{1}{2}\quad \text{or} \quad k=2$$
Answer
The value(s) of k for which the roots are real and equal is: $$\boxed{k=\frac{1}{2}\ \text{or}\ k=2}$$