Special Quadratics — Parametric, Logarithmic & Constructed

The roots of $x^2 + x + 1 = 0$ are the non-real cube roots of unity $\omega$ and $\omega^2$, with:

• $\omega^3 = 1$ (so powers cycle with period 3: $\omega^n = \omega^{\,n \bmod 3}$),
• $1 + \omega + \omega^2 = 0$,
• $\omega^2 = \bar{\omega} = \dfrac{1}{\omega}$ (the two roots are each other's square and reciprocal).

So if $x$ is a root of $x^2+x+1=0$: any $x^{3k} = 1$, and a sum like $x^n + x^{n+1} + x^{n+2} = x^n(1 + x + x^2) = 0$.

Two moves for symbol-coefficient quadratics:

• Sum-to-zero ⇒ unit root. For $(q-r)x^2 + (r-p)x + (p-q) = 0$, the coefficients add to $0$, so $x = 1$ is a root and the other root is the product $\dfrac{p-q}{q-r}$ (from $1 \cdot \beta = c/a$).
• Equal roots ⇒ a progression. A constructed equation whose roots are equal yields $D = 0$, which simplifies to an AP/GP/HP relation among the letters — often the HP form $\frac{2}{b^2} = \frac{1}{a^2} + \frac{1}{c^2}$ for the squared-coefficient family. The repeated root is $x = -\dfrac{B}{2A}$ (not $-B/A$).

Two strategies for an equation containing $|\,\cdot\,|$:

• Substitute the modulus: for $|x-a|^2 + |x-a| - 2 = 0$, set $t = |x-a| \ge 0$, solve $t^2 + t - 2 = 0$, keep only $t \ge 0$, then undo $t = |x-a|$.
• Split by sign: for $|f(x)| = g(x)$, first require $g(x) \ge 0$, then solve $f(x) = g(x)$ and $f(x) = -g(x)$, keeping only roots that satisfy the sign assumption.

Note $x^2 + k|x| + m = 0$ with $k, m > 0$ has no real root — a sum of non-negative terms can't vanish.

For a parametric quadratic at a given parameter value:

• Substitute the value, then factor. E.g. with $k = c$, $(a+b)x^2 - (a+b+c)x + c = 0$ factors as $[(a+b)x - c](x - 1) = 0$, giving $x = 1$ and $x = \dfrac{c}{a+b}$ directly — no formula needed.
• Minimum of a parametric quadratic: $x^2 + kx + k^2$ has minimum value at the vertex $= c - \dfrac{b^2}{4a} = k^2 - \dfrac{k^2}{4} = \dfrac{3k^2}{4}$.
• For a general parameter, the nature of the roots still comes from the discriminant in terms of that parameter.

If an equation is quadratic in $\log_b(\,\cdot\,)$, substitute $t = \log_b(\text{argument})$, solve the quadratic in $t$, then convert back with $t = \log_b u \iff u = b^{\,t}$. Useful log facts: $\log_b u = \dfrac{1}{\log_u b}$ (reciprocal of base and argument) and $\log_{b^2} u = \tfrac{1}{2}\log_b u$. When the coefficients of the quadratic are themselves logs, the same substitution turns Vieta's relations into relations among the logs.

Expand a constructed quadratic to standard form and read off Vieta's relations:

• $x^2 - ax - bx + (ab - c) = 0$ is $x^2 - (a+b)x + (ab-c) = 0$, so $\alpha+\beta = a+b$ and $\alpha\beta = ab - c$.
• To build the equation whose roots are a transform of $\alpha, \beta$ (their cubes, shifts, reciprocals), compute the new sum $S$ and product $P$ from $\alpha+\beta$ and $\alpha\beta$, then write $x^2 - Sx + P = 0$. E.g. if $\alpha^3, \beta^3$ are the new roots, $S = (\alpha+\beta)^3 - 3\alpha\beta(\alpha+\beta)$ and $P = (\alpha\beta)^3$.