Quadratic & Polynomial Equations

Everything for this chapter, in one place. $ax^2+bx+c=0$ , with $a \neq 0$ , roots $\alpha$ and $\beta$ .

Roots and the discriminant


Roots	$x = \dfrac{-b \pm \sqrt{D}}{2a}$
Discriminant	$D = b^2 - 4ac$

Nature of the roots — assumes $a,b,c$ are real

Condition	Roots
$D > 0$	two distinct real roots
$D = 0$	two equal real roots (one repeated)
$D < 0$	no real roots
$D > 0$ and a perfect square, $a,b,c$ rational	roots are rational
$D > 0$ , not a perfect square, $a,b,c$ rational	roots are irrational conjugates $p \pm \sqrt{q}$

Sum and product — use these instead of solving


$\alpha + \beta$	$-\dfrac{b}{a}$
$\alpha\beta$	$\dfrac{c}{a}$
$\alpha - \beta$	$\pm\dfrac{\sqrt{D}}{a}$
$(\alpha-\beta)^2$	$(\alpha+\beta)^2 - 4\alpha\beta$
$\alpha^2 + \beta^2$	$(\alpha+\beta)^2 - 2\alpha\beta$
$\alpha^3 + \beta^3$	$(\alpha+\beta)^3 - 3\alpha\beta(\alpha+\beta)$
$\dfrac{1}{\alpha} + \dfrac{1}{\beta}$	$\dfrac{\alpha+\beta}{\alpha\beta} = -\dfrac{b}{c}$
$\dfrac{1}{\alpha^2} + \dfrac{1}{\beta^2}$	$\dfrac{(\alpha+\beta)^2 - 2\alpha\beta}{(\alpha\beta)^2}$
Equation from its roots	$x^2 - (\alpha+\beta)x + \alpha\beta = 0$

Sign, maximum and minimum

Condition	Result
Vertex is at	$x = -\dfrac{b}{2a}$
Value there	$-\dfrac{D}{4a} = \dfrac{4ac-b^2}{4a}$
$a>0$	that value is the minimum; curve opens upward
$a<0$	that value is the maximum; curve opens downward
$a>0$ and $D<0$	$f(x) > 0$ for every real $x$
$a<0$ and $D<0$	$f(x) < 0$ for every real $x$

Common roots — for $a_1x^2+b_1x+c_1=0$ and $a_2x^2+b_2x+c_2=0$


Both roots shared	$\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}$
One root shared	subtract the scaled equations to eliminate $x^2$ , then solve for $x$

Cubic $ax^3+bx^2+cx+d=0$ with roots $\alpha,\beta,\gamma$


$\alpha+\beta+\gamma$	$-\dfrac{b}{a}$
$\alpha\beta + \beta\gamma + \gamma\alpha$	$\dfrac{c}{a}$
$\alpha\beta\gamma$	$-\dfrac{d}{a}$

Formula sheet