CBSE · Class 10 · Mathematics · Chapter 4

Quadratic Equations — Formula Sheet

Board Formulas

9 formulas

1.Standard Form
$a x^{2} + b x + c = 0, a \neq = 0$
Any degree-2 polynomial equation reduces to this form.
2.Quadratic Formula★
$x = \frac{- b \pm b ^{2} - 4 a c}{2 a}$
Gives both roots at once. Works even for irrational or complex roots.
3.Discriminant★
$D = b^{2} - 4 a c$
Decides the NATURE of roots — see next three rows.
4.Nature: Two Real Distinct Roots
$D > 0$
Graph crosses x-axis at two distinct points.
5.Nature: Two Equal Real Roots
$D = 0$
Graph touches x-axis at exactly one point. Roots = −b/(2a) (repeated).
6.Nature: No Real Roots
$D < 0$
Graph does not touch the x-axis. Roots are complex conjugates.
7.Sum and Product of Roots
$α + β = - \frac{b}{a}, α β = \frac{c}{a}$
Same as for the polynomial ax² + bx + c. Use to form a new equation with given roots.
8.Reconstructing a Quadratic from Roots
$x^{2} - (α + β) x + α β = 0$
Multiply by any non-zero scalar to get an equivalent equation.
9.Completing the Square Identity
$x^{2} + 2 p x + q = (x + p)^{2} + (q - p^{2})$
Basis of both the formula derivation and vertex-form of a parabola.