QUADRATICS

The toolkit

A quadratic equation is anything you can rewrite as

$ax^2 + bx + c = 0$

with $a \ne 0$. The three things you should be able to do quickly:

1. Factor, when the roots are integer-ish: $x^2 + 5x + 6 = (x+2)(x+3) = 0$.
2. Use the quadratic formula when factoring doesn't fall out:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.$

3. Use Vieta's formulas — without solving:
   • Sum of roots: $r_1 + r_2 = -\dfrac{b}{a}$
   • Product of roots: $r_1 r_2 = \dfrac{c}{a}$

Vieta is the single biggest time-saver on AMC10 quadratics. Most "find the sum/product of values of $x$" problems are one-liners with Vieta.

Worked example

If $x^2 + 5x + 6 = 0$, what is the sum of all distinct values of $x$?

By Vieta, the sum of roots is $-\dfrac{5}{1} = -5$. Done — no factoring needed.

When to slow down

If the problem asks for a specific root (not the sum/product), you usually need to factor or use the formula. If the discriminant $b^2 - 4ac$ is a perfect square, factoring will work.