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Class 10 · Chapter 4 · Mathematics

Quadratic Equations Cheat Sheet

Formulae

Standard form of a quadratic equation

ax2+bx+c=0,a≠0ax^{2} + bx + c = 0, \qquad a \neq 0

a, b and c are real numbers; a is the coefficient of x², b of x and c the constant term.

Quadratic formula

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}

Gives the roots of ax² + bx + c = 0 when the discriminant b² − 4ac is non-negative.

Discriminant

D=b2−4acD = b^{2} - 4ac

The sign of D decides the nature of the roots of ax² + bx + c = 0.

Sum and product of roots

α+β=−ba,αβ=ca\alpha + \beta = -\frac{b}{a}, \qquad \alpha\beta = \frac{c}{a}

For roots α and β of ax² + bx + c = 0.

Completing the square

(x+b2a)2=b2−4ac4a2\left(x + \frac{b}{2a}\right)^{2} = \frac{b^{2}-4ac}{4a^{2}}

Obtained by shifting the constant term and adding (b/2a)² to both sides.

Theorems

Nature of roots of a quadratic equation

For the equation ax² + bx + c = 0 with discriminant D = b² − 4ac: if D > 0 the roots are real and distinct; if D = 0 the roots are real and equal; if D < 0 there are no real roots.

D>0, D=0, D<0D > 0,\ D = 0,\ D < 0

When D < 0 the equation has no real roots (the roots are imaginary).

Definitions

Quadratic equation
An equation of the form ax² + bx + c = 0 where a ≠ 0; its degree is 2.
Root of a quadratic equation
A real number α such that aα² + bα + c = 0; also called a zero of the polynomial.
Discriminant
The quantity D = b² − 4ac, which determines the nature of the roots.
Real roots
Roots that are real numbers; they occur when D ≥ 0.
Equal roots
Two roots that have the same value; this happens when D = 0.
Factorisation method
Splitting the middle term to write the quadratic as a product of two linear factors.

Key Points

  • A quadratic equation has at most two roots.
  • A quadratic equation can be solved by factorisation, by completing the square, or by the quadratic formula.
  • Discriminant rule: D > 0 gives two distinct real roots; D = 0 gives two equal real roots; D < 0 gives no real roots.
  • If the roots are α and β, then α + β = −b/a and αβ = c/a.
  • A quadratic polynomial can be written as a(x − α)(x − β) using its roots.
  • If D = 0, each root equals −b/2a.
  • Word problems involving speed, time, area or consecutive numbers often reduce to quadratic equations.