Question

Check whether the following are quadratic equations: $(\mathrm{x}-3)(2\mathrm{x}+1)=\mathrm{x}(\mathrm{x}+5)$

Answer 1

Quadratic equation:

Quadratic equations are the polynomial equations of degree $2$ in one variable of type ${\mathrm{ax}}^2+\mathrm{bx}+\mathrm{c}$where $\mathrm{a},\mathrm{b},\mathrm{c},\in \mathrm{R}$ and$\mathrm{a}\ne 0$.

On expanding,

$\begin{array}{rcl}(\mathrm{x}-3)(2\mathrm{x}+1)& =& \mathrm{x}(\mathrm{x}+5)\\ 2{\mathrm{x}}^2+1\mathrm{x}-6\mathrm{x}-3& =& {\mathrm{x}}^2+5\mathrm{x}\\ 2{\mathrm{x}}^2-{\mathrm{x}}^2-5\mathrm{x}-5\mathrm{x}-3& =& 0\\ {\mathrm{x}}^2-10\mathrm{x}-3& =& 0\end{array}$

$\because$The above equation is in the form of ${\mathrm{ax}}^2+\mathrm{bx}+\mathrm{c}=0$, where $\mathrm{a}=1,\mathrm{b}=-10$ and $\mathrm{c}=-3$

Therefore, the given equation $(\mathrm{x}-3)(2\mathrm{x}+1)=\mathrm{x}(\mathrm{x}+5)$ is a quadratic equation.