Question

If the given expression $x^2-(5m-2)x+(4m^2+10m+25)$ can be expressed as a perfect square, then the value(s) of $m$ is/are

• A. $8$
• B. $-\frac{4}{3}$
• C. $\frac{4}{3}$
• D. $-8$

Answer 1

Answer: (A), (B)

$-\frac{4}{3}$

If the given expression $x^2-(5m-2)x+(4m^2+10m+25)$ can be rewritten as perfect square,
this means that the roots of equation $x^2-(5m-2)x+(4m^2+10m+25)=0$ will be equal.
$\Rightarrow D=0$
If we have $a{x}^2+bx+c=0$ then $D=b^2-4ac$
Here, we have $a=1,b=-(5m-2),c=(4m^2+10m+25)$
$\Rightarrow D=(5m-2{)}^2-4(4m^2+10m+25)=0\Rightarrow 25m^2-20m+4-16m^2-40m-100=0\Rightarrow 9m^2-60m-96=0\Rightarrow 3m^2-20m-32=0\Rightarrow (m-8\left)\right(3m+4)=0\therefore m=-\frac{4}{3},8$