# Show that the statement " If $x$ is real number such that $x^3+4x=0$, then $x$ is 0." is true using contra positive method.

Toolbox:
• Contrapositive method of proving a statement "if $p$ then $q$ is prove if ~$q$ then ~$p$.
The given statement is
" If $x$ is real number such that $x^3+4x=0$, then $x$ is 0."
Let $p :$ If $x$ is real number such that $x^3+4x=0.$
and
$q:$ $x=0$.
$\Rightarrow\:$ ~$p:$ $x^3+4x\neq 0$ $\forall\:x \in R$. and
~$q$: $x\neq 0$.
To show that the above statement is true using contra positive method,
Let us assume that $q$ is false.
We have to prove that $p$ is also false.
$i.e.,$ Let $x\neq 0$ and $x\in R$
$x^3+4x=0$
$\Rightarrow\:x(x^2+4)=0$
$\Rightarrow\:$ either $x=0$ or $x^2+4=0$
But our assumption is $x\neq 0$
$\Rightarrow\: x^2+4=0$
$x^2+4$ cannot be $0$ since $x^2 \geq 0\:\: \forall x \in R$
$\therefore$ $x^3+4x$ cannot be equal to zero
$\therefore$ The given statement is true.
edited Jul 26, 2014