Question

If 1 is a root of both the equation ax² + ax + 3 = 0 and x² + x + b = 0, then ab = _______.

Answer 1

Given:
ax² + ax + 3 = 0 ...(i)
x² + x + b = 0 ...(ii)

if x = 1 is the root of equation (i) and (ii), then it must satisfies the equations.

Putting x = 1 in (i), we get

$$\begin{gathered} a{\left(1\right)}^2+a\left(1\right)+3=0 \\ \Rightarrow a+a+3=0 \\ \Rightarrow 2a=-3 \\ \Rightarrow a=-\frac{3}{2} \end{gathered}$$

Putting x = 1 in (ii), we get
​
$$\begin{gathered} {\left(1\right)}^2+\left(1\right)+b=0 \\ \Rightarrow 1+1+b=0 \\ \Rightarrow b=-2 \end{gathered}$$

Therefore, $ab=\left(-\frac{3}{2}\right)\left(-2\right)=3$

Hence, ab = 3.