Question

If $a$ and $b$ are the roots of the equation $x^2+ax-b=0$, then find $a$ and $b$

Answer 1

Step 1: Express the quadratic equation in terms of sum and product of roots

The standard form of the quadratic equation with roots is given as,

$x^2-(\alpha +\beta )x+\alpha \beta =0$

where, $\alpha$ and $\beta$ are the roots of the equation.

Given, $x^2+ax-b=0$ and $a$ and $b$ are the roots of the equation.

Thus, here,

$\alpha =a$ and $\beta =b$

Step 2: Compare the given equation with standard form with roots

Comparing,

$$\begin{gathered} \Rightarrow \\ \Rightarrow \end{gathered}$$ $\begin{array}{rcl}-(\alpha +\beta )& =& a\\ -a-b& =& a[\because \alpha =a,\beta =b]\\ b& =& -2a\left(\mathrm{i}\right)\end{array}$

and,

$$\begin{gathered} \Rightarrow \\ \Rightarrow \\ \Rightarrow \end{gathered}$$ $\begin{array}{rcl}\alpha \beta & =& -b\\ ab& =& -b[\because \alpha =a,\beta =b]\\ a(-2a)& =& -(-2a)\\ a& =& -1\end{array}$

Substituting value of $a$ in $\left(\mathrm{i}\right)$,

$$\begin{gathered} \Rightarrow b=(-2)(-1) \\ \Rightarrow b=2 \end{gathered}$$

Therefore, the values of $a$ and $b$ are $-1$ and $2$ respectively.