Quadratic Equation formulas

Every formula JEE Advanced expects you to know for Quadratic Equation, with the variables spelled out. Free, no sign-up. Read it before a mock, not during one.

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Formulas: 26
Subject: Mathematics
Exam: JEE Advanced
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Quadratic Equation, every formula

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Quadratic formula
$x = \frac{- b \pm b ^{2} - 4 a c}{2 a}$
roots of ax^2+bx+c=0
Discriminant
$Δ = b^{2} - 4 a c$
Sum and product of roots
$α + β = - \frac{b}{a}, α β = \frac{c}{a}$
Difference of roots
$∣ α - β ∣ = \frac{b ^{2} - 4 a c}{∣ a ∣}$
Sum of squares of roots
$α^{2} + β^{2} = \frac{b ^{2} - 2 a c}{a ^{2}}$
Sum of cubes of roots
$α^{3} + β^{3} = \frac{3 ab c - b ^{3}}{a ^{3}}$
Factored form
$a x^{2} + b x + c = a (x - α) (x - β)$
Equation from its roots
$x^{2} - (α + β) x + α β = 0$
Nature of roots
$Δ > 0 : real, distinct; Δ = 0 : real, equal; Δ < 0 : complex$
a, b, c real
Root 1 when a+b+c=0
$a + b + c = 0 \Rightarrow one root = 1, other = \frac{c}{a}$
Root -1 when a-b+c=0
$a - b + c = 0 \Rightarrow one root = - 1, other = - \frac{c}{a}$
Roots in the ratio m : n
$(m + n)^{2} a c = mn b^{2}$
Roots differing by k
$b^{2} - 4 a c = a^{2} k^{2}$
Both roots common
$\frac{a _{1}}{a _{2}} = \frac{b _{1}}{b _{2}} = \frac{c _{1}}{c _{2}}$
for a_1x^2+b_1x+c_1=0 and a_2x^2+b_2x+c_2=0
Minimum of a quadratic (a>0)
$min = \frac{4 a c - b ^{2}}{4 a} at x = - \frac{b}{2 a}$
Maximum of a quadratic (a<0)
$max = \frac{4 a c - b ^{2}}{4 a} at x = - \frac{b}{2 a}$
Both roots greater than k
$Δ \geq 0, a f (k) > 0, k < - \frac{b}{2 a}$
f(x)=ax^2+bx+c
k lies between the roots
$a f (k) < 0$
Exactly one root in (k_1, k_2)
$f (k_{1}) f (k_{2}) < 0$
Cubic: relations between roots
$α + β + γ = - \frac{b}{a}, α β + β γ + γ α = \frac{c}{a}, α β γ = - \frac{d}{a}$
for ax^3+bx^2+cx+d=0
Roots-coefficients (degree n)
$\sum α = - \frac{a _{1}}{a _{0}}, \prod α = (- 1)^{n} \frac{a _{n}}{a _{0}}$
Remainder theorem
$f (x) \div (x - a) \Rightarrow remainder = f (a)$
Factor theorem
$f (a) = 0 ⟺ (x - a) is a factor of f (x)$
Conjugate complex roots
$real coefficients \Rightarrow complex roots occur as conjugate pairs$
Descartes' rule of signs
$# positive roots \leq # sign changes in f (x)$
Transform: reciprocal roots
$roots \frac{1}{α}, \frac{1}{β} \Rightarrow f (\frac{1}{x}) = 0$

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Quadratic Equation formulas

Quadratic Equation, every formula

Quadratic formula

Discriminant

Sum and product of roots

Difference of roots

Sum of squares of roots

Sum of cubes of roots

Factored form

Equation from its roots

Nature of roots

Root 1 when a+b+c=0

Root -1 when a-b+c=0

Roots in the ratio m : n

Roots differing by k

Both roots common

Minimum of a quadratic (a>0)

Maximum of a quadratic (a<0)

Both roots greater than k

k lies between the roots

Exactly one root in (k_1, k_2)

Cubic: relations between roots

Roots-coefficients (degree n)

Remainder theorem

Factor theorem

Conjugate complex roots

Descartes' rule of signs

Transform: reciprocal roots

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