Maharashtra State Board · Class 10 · Mathematics · Algebra Chapter 2

Quadratic Equations — Formula Sheet

Board Formulas

10 formulas

1.Standard Form★
$a x^{2} + b x + c = 0, a \neq = 0$
$a$ : Coefficient of x² (must be non-zero) · $b$ : Coefficient of x · $c$ : Constant term
Degree-2 polynomial equation. Real roots exist when D ≥ 0.
2.Quadratic Formula★
$x = \frac{- b \pm b ^{2} - 4 a c}{2 a}$
Works for every quadratic. Gives both roots at once.
3.Discriminant★
$D = b^{2} - 4 a c$
D > 0 two distinct real roots; D = 0 two equal real roots; D < 0 no real roots.
4.Sum of Roots
$α + β = - \frac{b}{a}$
Vieta's relation. Useful for verifying answers and forming new quadratics.
5.Product of Roots
$α β = \frac{c}{a}$
Together with the sum, uniquely determines both roots.
6.Forming Quadratic from Roots
$x^{2} - (α + β) x + α β = 0$
Take k = 1 for the simplest form. Multiply by any k ≠ 0 for a general quadratic.
7.Factorisation Method
$a x^{2} + b x + c = a (x - α) (x - β)$
Find two numbers that multiply to a·c and add to b — the splitting-the-middle-term trick.
8.Completing the Square
$x^{2} + p x + q = 0 \Rightarrow (x + p /2)^{2} = (p /2)^{2} - q$
Manual route to the quadratic formula. Works when factorisation fails.
9.Roots as Conjugate Surds
$If one root is p + q, other is p - q$
Applies when coefficients are rational and D is not a perfect square.
10.Relation Between Roots and Coefficients
$α^{2} + β^{2} = (α + β)^{2} - 2 α β$
Handy identity to compute α² + β² without solving the equation.