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Algebra

Quadratic Equations

ALL MODULES

CAT 2025 Lesson : Quadratic Equations - Concepts & Cheatsheet

Note: The video for this module contains a summary of all the concepts covered in this lesson. The video would serve as a good revision. Please watch this video in intervals of a few weeks so that you do not forget the concepts. Below is a cheatsheet that includes all the formulae but not necessarily the concepts covered in the video.

9. Cheatsheet

1

) A quadratic expression with

1

variable is of the form

ax^{2} + bx + c

, where

a \ne 0

.

2) Where

\alpha

and

\beta

are the roots, the quadratic equation formed is

(x - \alpha) (x - \beta) = 0

3

) Formula for roots:

x = \dfrac{-b \pm \sqrt{b^{2} - 4ac}}{2a}

, Discriminant:

D = b^{2} - 4ac

Discriminant	Nature of Roots
$D \gt 0$	Two real and distinct roots.
$D = 0$	Two real and equal roots
$D \lt 0$	Two imaginary roots

4)

For

f(x) = ax^{2} + bx + c

, Sum of roots

= \alpha + \beta = \dfrac{-b}{a}

and Product of roots

= \alpha \beta = \dfrac{c}{a}

5)

Changes to Roots: Where

\alpha

and

\beta

are the roots of

ax^{2} + bx + c = 0

, and

k

is a constant,

For the roots	The quadratic equation is
$(\alpha + k)$ and $(\beta + k)$	$a(x - k)^{2} + b(x - k) + c = 0$
$(\alpha - k)$ and $(\beta - k)$	$a(x + k)^{2} + b(x + k) + c = 0$
$(\alpha k)$ and $(\beta k)$	$a \left(\dfrac{x}{k} \right)^{2} + b \left(\dfrac{x}{k} \right) + c = 0$
$\left(\dfrac{\alpha}{k} \right)$ and $\left(\dfrac{\beta}{k} \right)$	$a(xk)^{2} + b(xk) + c = 0$
$\left(\dfrac{1}{\alpha} \right)$ and $\left(\dfrac{1}{\beta} \right)$	$cx^{2} + bx + a = 0$

Ratio of Roots: For a quadratic equation

\bm{ax^{2} + bx + c = 0}

, if the ratio of roots, i.e.,

\bm{\alpha : \beta = p : q,}

then

\bm{b^{2}pq = ac (p + q)^{2}}

7)

Cubic Equation: If

\alpha, \beta

and

\gamma

are the roots of the equation

ax^{3} + bx^{2} + cx + d = 0

, then

\alpha + \beta + \gamma = \dfrac{-b}{a}

\alpha \beta + \beta \gamma + \gamma \alpha = \dfrac{c}{a}

\alpha \beta \gamma = \dfrac{-d}{a}

8)

Biquadratic Equation: If

\alpha, \beta, \gamma

and

\delta

are the roots of the equation

ax^{4} + bx^{3} + cx^{2} + dx + e = 0

\alpha + \beta + \gamma + \delta = \dfrac{-b}{a}

\alpha \beta + \alpha \gamma + \alpha \delta + \beta \gamma + \beta \delta + \gamma \delta = \dfrac{c}{a}

\alpha \beta \gamma + \alpha \beta \delta + \alpha \gamma \delta + \beta \gamma \delta = \dfrac{-d}{a}

\alpha \beta \gamma \delta = \dfrac{e}{a}

9)

Descarte's rule: Number of sign changes in

f(x)

and

f(-x)

are the maximum number of positive and negative real roots of a polynomial. These numbers vary by a multiple of

2

10)

Minimum or Maximum value of

f(x) = \dfrac{(4ac - b^{2})}{4a}

x = \dfrac{-b}{2a}

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