Question

For what value of $k$, are the roots of the quadratic equation $(k+4)x^2+(k+1)x+1=0$ equal?

Answer 1

Step 1: Estimate the constants of the equation

The given equation is $(k+4)x^2+(k+1)x+1=0$.

For equal roots,

$b^2-4ac=0$

In this case $a=k+4,b=k+1,c=1$.

Step 2: Calculate the value of $k$

Hence,

$b^2-4ac=0$

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

$\Rightarrow$$k^2+2k+1-4k-16=0$

$\Rightarrow$ $k^2-2k+1=16$

$\Rightarrow$ ${\left(k-1\right)}^2=4^2$

$\Rightarrow$ $k-1=\pm 4$

$\Rightarrow$ $k=4-1,-4-1$

$\Rightarrow$ $k=3,-5$

Hence, the required values of $k$ are $3$ and $-5$.