Question

If $f\left(x\right)=4x^2+3x-7$ and $\alpha$ is a common root of the equation $x^2-3x+2=0$ and $x^2+2x-3=0$ then find the value of $f\left(\alpha \right)$.

Answer 1

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

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

$\Rightarrow (x-1)(x-2)=0,$ Roots $=1,2$

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

$\Rightarrow x^2+3x-x-3=0$

$\Rightarrow (x+3)(x-1)=0$ Roots $=1,-3$

common root $=1$

$\Rightarrow f\left(1\right)=4{\left(1\right)}^2+3\left(1\right)-7$

$=7-7$

$=0$

$1$ is a root of $f\left(x\right)$ as well.

Hence, the answer is $0.$