Quadratic Equations

Quadratic Equations

A quadratic $ax^2+bx+c=0$ is the most-tested single object in CAT algebra. You almost never need the full formula — Vieta's relations and the discriminant answer most questions directly.

The two tools

$\text{sum of roots}=\alpha+\beta=-\frac{b}{a},\qquad \text{product}=\alpha\beta=\frac{c}{a}.$

$\text{Discriminant}\ D=b^2-4ac:\quad D>0\ \text{(two real)},\ \ D=0\ \text{(equal)},\ \ D<0\ \text{(none real)}.$

Build a quadratic from its roots: $x^2-(\alpha+\beta)x+\alpha\beta=0$.

A worked example

The roots of a quadratic satisfy $\alpha+\beta=5$ and $\alpha\beta=6$. Find $\alpha^2+\beta^2$.

Use the identity $\alpha^2+\beta^2=(\alpha+\beta)^2-2\alpha\beta$ — no need to find the roots:

$\alpha^2+\beta^2=5^2-2(6)=25-12=\mathbf{13}.$

(The roots happen to be $2$ and $3$, but Vieta got the answer without solving.)

Handy symmetric identities

• $\alpha^2+\beta^2=(\alpha+\beta)^2-2\alpha\beta$
• $\dfrac1\alpha+\dfrac1\beta=\dfrac{\alpha+\beta}{\alpha\beta}$
• $(\alpha-\beta)^2=(\alpha+\beta)^2-4\alpha\beta=\dfrac{D}{a^2}$

Common traps

• Sign of $b$. Sum of roots is $-b/a$ — the minus sign trips people constantly.
• "Real and distinct" vs "real". Distinct needs $D>0$ strictly; $D=0$ is real but repeated.
• Maximum/minimum. The vertex value $-\dfrac{D}{4a}$ gives the extreme of $ax^2+bx+c$.

Checklist

• Reach for sum/product before the formula
• Read $D$ for the nature of the roots
• Express the target via symmetric identities
• Watch the minus sign in $-b/a$