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Quadratic Equations & Polynomial Inequalities formula sheet

Every formula in the Quadratic Equations & Polynomial Inequalities chapter for JEE Main, with one line on when it applies. Topic 1 is open to everyone; sign in free to read all 24 and download the PDF.

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How we count: 7× asked means 7 past JEE Main questions in the PrepWiser bank needed that formula, and "last" is the latest year one appeared. ★ marks formulas asked 8 or more times.

Roots and their nature

Quadratic formula
8× asked, last 2026
x=−b±D2a,D=b2−4acx=\frac{-b\pm\sqrt D}{2a},\qquad D=b^2-4ac
Real a,b,ca,b,c: D>0D>0 two real, D=0D=0 equal (−b/2a-b/2a), D<0D<0 conjugate pair. a+b+c=0a+b+c=0: roots 1 and c/ac/a.
Rational and integer roots
3× asked, last 2025
a,b,c∈Q: roots rational  ⟺  D is a perfect squarea,b,c\in\mathbb Q:\ \text{roots rational}\iff D\text{ is a perfect square}
x2+bx+cx^2+bx+c (integers): integer roots iff DD is a perfect square.
Conjugate roots
4× asked, last 2025
p+iq root⇒p−iq rootp+q root⇒p−q root\begin{aligned}&p+iq\text{ root}\Rightarrow p-iq\text{ root}\\&p+\sqrt q\text{ root}\Rightarrow p-\sqrt q\text{ root}\end{aligned}
First needs real, second rational coefficients. x2+kx+k2=0x^2+kx+k^2=0: roots kωk\omega, kω2k\omega^2.
Sum of squares = 0
4× asked, last 2024
(px−q)2+(qx−r)2=0 ⇒ x=qp=rq(px-q)^2+(qx-r)^2=0\ \Rightarrow\ x=\frac qp=\frac rq
Real terms: each square is 0, so p,q,rp,q,r in G.P. with ratio xx.

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Practise Quadratic Equations & Polynomial Inequalities

20 more formulas in 5 topics

Sum, product and symmetric functions; Forming equations and common roots; Sign, range and location of roots; Equations reducible to quadratics; and Modulus, logarithms and inequalities. Read the first part free. Sign in free with Google to read the rest and download the PDF.

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Sum, product and symmetric functions

  • Sum and product14×
  • Symmetric functions13×
  • Power sums9×
  • Cubic: Vieta5×

Forming equations and common roots

  • Equation with given roots5×
  • Transformed roots3×
  • One common root8×
  • Both roots common1×

Sign, range and location of roots

  • Sign of a quadratic3×
  • Maximum and minimum7×
  • Range of a quadratic ratio2×
  • Signs of the roots5×
  • Roots relative to k4×

Equations reducible to quadratics

  • Substitution ranges13×
  • Reciprocal equations8×
  • Higher degree8×
  • Square-root equations3×

Modulus, logarithms and inequalities

  • Modulus14×
  • Wavy curve4×
  • Logarithms4×

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About 125 past JEE Main questions from this chapter, with instant answer checking and step-by-step solutions where available. Free, and you can start without signing up.

Practise Quadratic Equations & Polynomial Inequalities

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