Question

For each equation, given below, find the value of ′m′ so that equation has equal roots. Also, find the solution of each equation.

(ii). $3x^2+12x+(m+7)=0$

Answer 1

Given equation: $3x^2+12x+(m+7)=0$

For equal roots, Discriminant $\left(D\right)=0$ where $D=b^2-4ac$

On comparing $3x^2+12x+(m+7)=0$ with $a{x}^2+bx+c=0$, we get $a=3,b=12\text{and}c=m+7$

$D=(12{)}^2-4\times 3(m+7)=0$

$\Rightarrow 144-12m-84=0$

$\Rightarrow m=\frac{60}{12}=5$

Putting value of m in given equation, we get

$3x^2+12x+12=0$

$\Rightarrow x^2+4x+4=0$

$\Rightarrow (x+2{)}^2=0$ $\{\because a^2+2ab+b^2=(a+b{)}^2\}$

$\Rightarrow x+2=0$

$\Rightarrow x=-2$

$\therefore m=5$ and $x=-2$.