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Formula Sheets / Maths

Chapter 2

Quadratic Equations

Revise with flashcards Download this chapter (PDF) All Maths formulas (PDF)

Roots and coefficients Nature of roots Common roots Location of roots Sign of a quadratic

Roots and coefficients

Roots
$a x^{2} + b x + c = 0 \Rightarrow x = \frac{- b \pm D}{2 a},$ $D = b^{2} - 4 a c$
Sum and product
$∣ α - β ∣ = \frac{D}{∣ a ∣}$ $α β = \frac{c}{a},$ $∣ α - β ∣ = \frac{D}{∣ a ∣}$
Equation from roots
$x^{2} - (α + β) x + α β = 0$
Cubic
$a x^{3} + b x^{2} + c x + d = 0 : \sum α = - \frac{b}{a}, \sum α β = \frac{c}{a}, α β γ = - \frac{d}{a}$

Nature of roots

$D > 0 :$ real, distinct $;$ $D = 0 :$ real, equal $;$ $D < 0 :$ non-real (conjugate pair)
Rational coefficients
$D$ a perfect square $\Rightarrow$ rational roots $;$ irrational roots occur in pairs $p \pm q$

Common roots

One common root
$(c_{1} a_{2} - c_{2} a_{1})^{2} = (a_{1} b_{2} - a_{2} b_{1}) (b_{1} c_{2} - b_{2} c_{1})$
Both roots common
$\frac{a _{1}}{a _{2}} = \frac{b _{1}}{b _{2}} = \frac{c _{1}}{c _{2}}$

Location of roots

Vertex
$f (x) = a x^{2} + b x + c :$ vertex at $x = - \frac{b}{2 a},$ $f_{m i n / m a x} = - \frac{D}{4 a}$
Both roots greater than k
$D \geq 0,$ $a f (k) > 0,$ $- \frac{b}{2 a} > k$
Both roots less than k
$D \geq 0,$ $a f (k) > 0,$ $- \frac{b}{2 a} < k$
k lies between the roots
$D \geq 0, a f (k_{1}) > 0, a f (k_{2}) > 0, k_{1} < - \frac{b}{2 a} < k_{2}$
Exactly one root in (k₁, k₂)
$f (k_{1}) f (k_{2}) < 0$
Both roots in (k₁, k₂)
$D \geq 0, a f (k_{1}) > 0, a f (k_{2}) > 0, k_{1} < - \frac{b}{2 a} < k_{2}$

Sign of a quadratic

$a x^{2} + b x + c > 0 \forall x \in R ⟺ a > 0, D < 0;$ $< 0 \forall x ⟺ a < 0, D < 0$

Basic Mathematics and Logarithm Sequence and Series