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Quadratic Equations & Polynomial Inequalities revision notes

Every idea in the Quadratic Equations & Polynomial Inequalities chapter for JEE Main, with the usual mistakes and the fast methods. 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 concept, and "last" is the latest year one appeared. ★ marks ideas asked 8 or more times.

Roots and their nature

Quadratic formula and discriminant
8× asked, last 2026
x=−b±D2a,D=b2−4acx=\frac{-b\pm\sqrt{D}}{2a},\qquad D=b^2-4ac
a≠0a\ne0; a,b,ca,b,c real. D>0D>0: two distinct real roots; D=0D=0: equal roots −b/2a-b/2a; D<0D<0: non-real conjugate pair.
xxyy
D>0D>0: x2−2x−3x^2-2x-3
xxyy
D=0D=0: x2−2x+1x^2-2x+1
xxyy
D<0D<0: x2−2x+3x^2-2x+3
  • If a+b+c=0a+b+c=0, then x=1x=1 is a root and the other is c/ac/a. So a(b−c)x2+b(c−a)x+c(a−b)=0a(b-c)x^2+b(c-a)x+c(a-b)=0 has equal roots only when both are 1: c(a−b)=a(b−c)c(a-b)=a(b-c), i.e. b=2ac/(a+c)b=2ac/(a+c).
  • Not D>0D>0 for "distinct roots". A non-real conjugate pair is also distinct, so only D≠0D\ne0 is needed unless the roots must be real.
Rational and integer roots
3× asked, last 2025
a,b,c∈Q: roots rational  ⟺  D is the square of a rationala,b,c\in\mathbb{Q}:\ \text{roots rational}\iff D\ \text{is the square of a rational}
For x2+bx+cx^2+bx+c with integers b,cb,c: integer roots iff DD is a perfect square (D\sqrt D and bb then share parity).
  • x2+4x−n=0x^2+4x-n=0 has D=4(n+4)D=4(n+4), so integer roots need n+4n+4 to be a perfect square. For n∈[20,100]n\in[20,100] count squares in [24,104][24,104]: 25 to 100, six values.
Conjugate roots
4× asked, last 2025
p+iq root⇒p−iq root,p+q root⇒p−q rootp+iq\ \text{root}\Rightarrow p-iq\ \text{root},\qquad p+\sqrt q\ \text{root}\Rightarrow p-\sqrt q\ \text{root}
First needs real coefficients; second needs rational coefficients (q>0q>0 not a perfect square).
  • Quick method. x2+kx+k2=0x^2+kx+k^2=0 has roots kω,kω2k\omega,k\omega^2, so α3=β3=k3\alpha^3=\beta^3=k^3; x2−kx+k2=0x^2-kx+k^2=0 has −kω,−kω2-k\omega,-k\omega^2, cube −k3-k^3. Use these to cut huge powers.
Sum of squares equal to zero
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 x,p,q,rx,p,q,r: each square is 0, so p,q,rp,q,r are in G.P. with ratio xx. Any sum of squares or moduli is 0 only if each term is.
  • Triangle sides a,ar,ar2a,ar,ar^2 need r2<1+rr^2<1+r and r+r2>1r+r^2>1, so r∈((5−1)/2,(5+1)/2)r\in((\sqrt5-1)/2,(\sqrt5+1)/2). "Not all of the same sign" forces a negative ratio.

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Practise Quadratic Equations & Polynomial Inequalities

20 more concepts 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 product of roots14×
  • Symmetric functions of two roots13×
  • Power sums by recurrence9×
  • Cubic and quartic: 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 quadratic expression3×
  • Maximum and minimum value7×
  • Range of a quadratic ratio2×
  • Signs of the roots5×
  • Roots relative to a number k4×

Equations reducible to quadratics

  • Substitution and the allowed range13×
  • Reciprocal equations8×
  • Higher-degree polynomials8×
  • Equations with square roots3×

Modulus, logarithms and inequalities

  • Modulus equations14×
  • Wavy curve (sign scheme)4×
  • Logarithm rules and domain4×

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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.

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