Mathematics - Quadratic Equation Question with Solution | TestHub

Let $x=ab+bc+ca$.

From $a+2b+c=4$, we have $a=4-2b-c$.

$x=-2b^2+4b-2bc+4c-c^2$

Rearrange this as a quadratic equation in $b$:

$2b^2+(2c-4)b+(c^2-4c+x)=0$

Since $b$ is a real number, the discriminant of this quadratic equation must be non-negative.

$\Delta =(2c-4{)}^2-4(2)(c^2-4c+x)\ge 0$

$c^2-4c-4+2x\le 0$

Now, consider this as a quadratic inequality in $c$

${\Delta }_c=(-4{)}^2-4(1)(-4+2x)\ge 0$

$x\le 4$