Question

How many real roots does the quadratic equation $f\left(x\right)=x^2+3\left|x\right|+2=0$ have?

• A. One
• B. Two
• C. Four
• D. No real root

Answer 1

Answer: (D)

No real root

$|x|=x$ when $x>0$

$|x|=-x$ when $x<0$

So when $x>0$:

$f\left(x\right)=x^2+3x+2=0$

$\Rightarrow (x+1)(x+2)=0$

$\Rightarrow x=-1,-2$

So, no root exists which is $>0$

When $x<0$:

$f\left(x\right)=x^2-3x+2=0$

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

$\Rightarrow x=1,2$

So, no root exists which is $<0$

Thus, no real root exists for the function $f\left(x\right)=x^2+3|x|+2=0$