Algebra — common mistakes

For inequalities like $|f(x)| > |g(x)|$, squaring both sides is usually the safest method. It avoids the confusion of setting up multiple positive/negative cases. Remember that for any real number $a$, $|a|^2 = a^2$.

A very common mistake is forgetting to factor out the constant term before expanding, or forgetting to apply the power to that constant (like the $4^{1/2}=2$ in the example). Always rewrite $(a+bx)^n$ as $a^n(1+\frac{b}{a}x)^n$ first.

Graphically, the V-shape of $|f(x)|$ intersects the horizontal line $y=k$ at two points. The inequality holds for the parts of the 'V' that are above the line, which are the two 'arms' extending outwards, leading to two separate regions.

It means a quadratic expression that cannot be factorised using real numbers. This occurs when its discriminant, $b^2-4ac$, is negative. For example, in $x^2+4$, the discriminant is $0^2 - 4(1)(4) = -16 < 0$. Its roots are complex.

The expansion is an infinite geometric series in disguise. For an infinite series to converge to a finite sum, the common ratio must have a magnitude less than 1. In the binomial expansion, the terms behave like a geometric progression with ratio related to $x$. If $|x| \ge 1$, the terms get larger and larger, so the sum diverges to infinity and is not a useful approximation.

If $x=2$ is a root, then $(x-2)$ is a factor. You can use polynomial long division to divide the original cubic by $(x-2)$. The result will be a quadratic expression, which you can then solve using the quadratic formula or factorisation to find the remaining two roots.