36. $x^4-2x^2-63\le 0$

Expert intro: Treat the expression as a quadratic in $x^2$. This turns a fourth-degree inequality into a familiar factoring problem.

Detailed walkthrough

Let

$t=x^2$

Then

$x^4-2x^2-63\le 0$

becomes

$t^2-2t-63\le 0$

Factor:

$t^2-2t-63=(t-9)(t+7)$

So we solve

$(t-9)(t+7)\le 0$

The roots are $t=-7$ and $t=9$. Since the parabola opens upward, the expression is nonpositive between the roots:

$-7\le t\le 9$

Now use $t=x^2\ge 0$, so the valid part is

$0\le x^2\le 9$

Thus

$-3\le x\le 3$

Final answer:

$[-3,3]$

💡 Pitfall guide

A frequent mistake is to keep the interval $[-7,9]$ for $t$ and forget that $t=x^2$ cannot be negative. The condition $x^2\ge 0$ cuts off the negative part automatically.

🔄 Real-world variant

If the inequality were $x^4-2x^2-63<0$, the solution would still be

$(-3,3)$

because the endpoints correspond to $x^2=9$, where the expression becomes $0$. If the inequality were $\ge 0$, the solution would be

$(-\infty,-3]\cup[3,\infty)$

🔍 Related terms

quadratic in x^2, quartic equation, interval solution