Identities

$$(a + b)^2 = a^2 + 2ab + b^2 \qquad (a - b)^2 = a^2 - 2ab + b^2 \qquad a^2 - b^2 = (a - b)(a + b)$$ $$a^3 + b^3 = (a + b)(a^2 - ab + b^2) \qquad a^3 - b^3 = (a - b)(a^2 + ab + b^2)$$

Quadratic equation $ax^2 + bx + c = 0$

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

• Discriminant $\Delta = b^2 - 4ac$: $\Delta > 0$ two real roots, $\Delta = 0$ one repeated root, $\Delta < 0$ no real roots.
• Sum of roots $= -\dfrac{b}{a}$, product of roots $= \dfrac{c}{a}$.
• If the roots are $\alpha, \beta$: $x^2 - (\alpha + \beta)x + \alpha\beta = 0$.

Completing the square

$$x^2 + bx + c = \left(x + \frac{b}{2}\right)^2 + c - \frac{b^2}{4}$$ Turning point of $y = ax^2 + bx + c$ is at $x = -\dfrac{b}{2a}$.

Factorising $ax^2 + bx + c$

Find two numbers with product $ac$ and sum $b$, split the middle term, then factorise by grouping.