IIT JEE Test Series — Practice smarter, perform stronger.

Quadratic Equation formulas

Master Quadratic Equation through 26 JEE Advanced-level formulas, systematically structured with every variable spelled out. Revise concept-wise, identify the areas where you need improvement, and focus your preparation with greater precision.

Practise these formulas Know more about IIT JEE

Back to Formula Book

Quadratic Equation, every formula

26 formulas, typeset and free. Print it, or keep it open beside your practice.

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
$a x^{2} + b x + c = a (x - α) (x - β)$
Q1MCQFactored form
$x^{2}-5x+6$ factors as:
- A
  $(x-2)(x-3)$
- B
  $(x+2)(x+3)$
- C
  $(x-1)(x-6)$
- D
  $(x-2)(x+3)$
Equation from its roots
$x^{2} - (α + β) x + α β = 0$
Q1MCQEquation from roots
The quadratic equation whose roots are $2$ and $3$ is:
- A
  $x^{2}-5x+6=0$
- B
  $x^{2}+5x+6=0$
- C
  $x^{2}-6x+5=0$
- D
  $x^{2}-x-6=0$
Nature of roots
$Δ > 0 : real, distinct; Δ = 0 : real, equal; Δ < 0 : complex$
a, b, c real
Q1MCQNature of roots
The roots of $x^{2}+x+1=0$ are:
- A
  complex (non-real)
- B
  real and distinct
- C
  real and equal
- D
  rational
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
Q1MCQCommon roots
The equations $x^{2}-5x+6=0$ and $2x^{2}-10x+12=0$ have:
- A
  both roots common
- B
  exactly one common root
- C
  no common root
- D
  complex roots
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
Q1MCQBoth roots > k
For both roots of $f(x)=x^{2}+bx+c$ to be greater than a number $k$, a necessary condition (with $a=1>0$) is:
- A
  $f(k)>0$
- B
  $f(k)<0$
- C
  $f(k)=0$
- D
  $\Delta<0$
k lies between the roots
$a f (k) < 0$
Q1MCQk between roots
If a number $k$ lies between the roots of $f(x)=ax^{2}+bx+c$, then:
- A
  $a\,f(k)<0$
- B
  $a\,f(k)>0$
- C
  $f(k)=0$
- D
  $\Delta<0$
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)
$\sum α = - \frac{a _{1}}{a _{0}}, \prod α = (- 1)^{n} \frac{a _{n}}{a _{0}}$
Q1MCQProduct of roots (degree n)
For $a_0x^{n}+a_1x^{n-1}+\cdots+a_n=0$, the product of all roots equals:
- A
  $(-1)^{n}\dfrac{a_n}{a_0}$
- B
  $\dfrac{a_n}{a_0}$
- C
  $-\dfrac{a_1}{a_0}$
- D
  $(-1)^{n}\dfrac{a_0}{a_n}$
Remainder theorem
$f (x) \div (x - a) \Rightarrow remainder = f (a)$
Factor theorem
$f (a) = 0 ⟺ (x - a) is a factor of f (x)$
Q1MCQFactor theorem
$(x-2)$ is a factor of a polynomial $f(x)$ if and only if:
- A
  $f(2)=0$
- B
  $f(0)=2$
- C
  $f'(2)=0$
- D
  $f(-2)=0$
Conjugate complex roots
$real coefficients \Rightarrow complex roots occur as conjugate pairs$
Q1MCQConjugate roots
If $2+3i$ is a root of a polynomial with real coefficients, another root must be:
- A
  $2-3i$
- B
  $-2+3i$
- C
  $3+2i$
- D
  $-2-3i$
Descartes' rule of signs
$# positive roots \leq # sign changes in f (x)$
Transform: reciprocal roots
$roots \frac{1}{α}, \frac{1}{β} \Rightarrow f (\frac{1}{x}) = 0$
Q1MCQReciprocal roots
The equation whose roots are the reciprocals of the roots of $f(x)=0$ is:
- A
  $f\!\left(\tfrac1x\right)=0$
- B
  $f(-x)=0$
- C
  $f(x-1)=0$
- D
  $f(2x)=0$

More JEE Advanced Mathematics formula sheets

Every chapter sheet is typeset, free and open without a sign-in.

Other ways to revise this chapter

Master this chapter with similar other learning materials.

Preparing students for India’s top institutes

Our students are currently into top technological and medical institutes of India.

IIT Bombay
IIT Delhi
IIT Madras
IIT Kanpur
IIT Kharagpur
IIT Roorkee
IIT Guwahati
IIT BHU Varanasi
AIIMS Delhi
NIT Tiruchirappalli
NIT Rourkela
IIT Bombay
IIT Delhi
IIT Madras
IIT Kanpur
IIT Kharagpur
IIT Roorkee
IIT Guwahati
IIT BHU Varanasi
AIIMS Delhi
NIT Tiruchirappalli
NIT Rourkela

Join QuestPix, Today!

Get notified first, with exam & curriculum updates, course & test series launch offers, motivation & success stories and free learning resources recommended by toppers.

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

More JEE Advanced Mathematics formula sheets

Other ways to revise this chapter

Preparing students for India’s top institutes

Join QuestPix, Today!