Quadratic Formula

Completing the square. Any quadratic $P(x)=ax^2+bx+c$ can be rewritten as $a\left(x+\dfrac b{2a}\right)^2+P\left(-\dfrac b{2a}\right)$ -- verified by expanding the bracket and simplifying. Setting $P(x)=0$ and solving for $x$ from this form gives the quadratic formula:

The discriminant $D=b^2-4ac$ governs the nature of the roots:

When $D<0$, the two roots are the complex pair $\alpha=\dfrac{-b+i\sqrt{4ac-b^2}}{2a},\ \beta=\dfrac{-b-i\sqrt{4ac-b^2}}{2a}$ (with $i^2=-1$), studied fully in a later class.

Sum and product of roots. If $\alpha,\beta$ are the roots of $ax^2+bx+c=0$, then

$\alpha+\beta=-\dfrac ba,\qquad \alpha\beta=\dfrac ca.$ …