Quadratic Equations & Nature of Roots

At a glance 11 formulas

1. 1Quadratic formula$x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$\text{roots of } ax^2 + bx + c = 0,\ a \ne 0$
2. 2Nature of roots$D = b^2 - 4ac \text{ for } ax^2 + bx + c = 0$$D > 0: \text{two real};\ D = 0: \text{equal};\ D < 0: \text{no real roots}$
3. 3Sum and product$\alpha + \beta = -\dfrac{b}{a},\ \alpha\beta = \dfrac{c}{a}$$\alpha, \beta \text{ roots of } ax^2 + bx + c = 0$
4. 4Symmetric expressions$\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta$$(\alpha - \beta)^2 = (\alpha + \beta)^2 - 4\alpha\beta;\ \alpha, \beta \text{ the roots}$
5. 5Equation from its roots$x^2 - Sx + P = 0$$S = \text{sum},\ P = \text{product of the roots};\ \text{roots } \tfrac{1}{\alpha}, \tfrac{1}{\beta}:\ cx^2 + bx + a = 0$
6. 6Signs of the rootsWith sum $S$ and product $P$: $P < 0$ means opposite signs; $P > 0$ and $S > 0$, both positive; $P > 0$ and $S < 0$, both negative.
7. 7Miscopied coefficientA wrong constant still gives the right sum of roots; a wrong $x$-coefficient still gives the right product. Build the equation from the two survivors.
8. 8Common rootSubtract the two equations to remove $x^2$; the common root comes out in terms of the parameter. Put it back into either equation.
9. 9Integer rootsInteger roots $r, s$ of $x^2 + bx + c$: factor pairs of $c$ (signs too) with sum $-b$. If $k$ is in two coefficients, eliminate it: $rs = 2(r + s)$ becomes $(r - 2)(s - 2) = 4$.
10. 10Pair the four factorsIn $(x - p)(x - q)(x - r)(x - s)$ with $p + s = q + r$, multiply the outer pair and the inner pair: both start $x^2 - (p + s)x$, so call that part $y$.
11. 11Check the storyA word problem gives two roots; keep only the one the story allows, never a negative count, length, price or speed.