General form of Quadratic Equation:

PYQ-2023-Quadratic_Equation-Q2, PYQ-2023-Quadratic_Equation-Q3, PYQ-2023-Quadratic_Equation-Q8, PYQ-2023-Quadratic_Equation-Q9, PYQ-2023-Probability-Q5, PYQ-2023-Probability-Q13, PYQ-2023-Statistics-Q8, PYQ-2023-Trigonometric_Ratios-Q4, PYQ-2023-AOD-Q5

$$ a x^{2}+b x+c=0, a \neq 0 $$

$\quad \quad $ where $a$, $b$, $c$ are constants

• Roots of equations $$\alpha=\frac{-b+\sqrt{b^{2}-4 a c}}{2 a}, \beta=\frac{-b-\sqrt{b^{2}-4 a c}}{2 a}$$

Sum and Product of Roots:

PYQ-2023-Quadratic_Equation-Q5, PYQ-2023-Quadratic_Equation-Q6

$\quad$ If $\alpha$ and $\beta$ are the roots of the quadratic equation $a x^{2}+b x+c=0$, then

• Sum of roots $$\alpha+\beta=-\frac{b}{a}$$

• Product of roots $$\alpha \beta=\frac{c}{a}$$

Discriminant of Quadratic equation:

$\quad $ The Discriminant of the quadratic equation $a x^{2}+b x+c=0$ is given by $D=b^{2}-4 a c$.

Nature of Roots:

PYQ-2023-Quadratic_Equation-Q4, PYQ-2023-Area_Under_Curves-Q8, PYQ-2023-Function-Q19

• If $D=0$, the roots are real and equal $\alpha=\beta=-\frac{b}{2 a}$.

• If $D > 0$, The roots are real and unequal.

• If $D<0$, the roots are imaginary and unequal.

• If $D>0$ and $D$ is a perfect square, the roots are rational and unequal.

• If $D>0$ and $D$ is not a perfect square, the roots are irrational and unequal.

Formation of Quadratic Equation with given roots

$\quad $ If $\alpha$ and $\beta$ are the roots of the quadratic equation, then

$$(x-\alpha)(x-\beta)=0 \ \text{or} \ x^{2}-(\alpha+\beta) x+\alpha \beta=0 $$

$$ x^{2}-(\ \text{Sum of roots } ) \ x+ \ \text{product of roots} =0$$

Common Roots:

PYQ-2023-Quadratic_Equation-Q7, PYQ-2023-AOD-Q7

• If two quadratic equations $a_{1} x^{2}+b_{1} x+c_{1}=0 $ $\ \text{and} \ a_{2} x^{2}+b_{2} x+c_{2}=0$ have both roots common, then $\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}$.

• If only one root $\alpha$ is common, then

$$\alpha = \frac{c_{1} a_{2} - c_{2} a_{1}}{a_{1} b_{2} - a_{2} b_{1}} = \frac{b_{1} c_{2} - b_{2} c_{1}}{c_{1} a_{2} - c_{2}}$$

Range of Quadratic Expression:

$f(x) = ax^2 + bx + c$ in restricted domain $x \in [x_1, x_2]$

• If $-\frac{b}{2a} \notin [x_1, x_2]$, then

$$ f(x) \in \left[\min\left({f(x_1), f(x_2)}\right), \max\left({f(x_1), f(x_2)}\right)\right] $$

• If $-\frac{b}{2a} \in [x_1, x_2]$, then

$$f(x) \in \left[\min\left({(f(x_1), f(x_2)), -\frac{D}{4a}}\right), \max\left({(f(x_1), f(x_2)), -\frac{D}{4a}}\right )\right]$$

Location of Roots:

$\quad$ Let $f(x)=a x^{2}+b x+c$, where $a>0$ & $a \cdot b \cdot c \in R$.

• Conditions for both the roots of $f(x)=0$ to be greater than a specified number’ $x_{0}$ ’ are $b^2 - 4 a c \geq 0;$ $ f(x_0)>0 $ & $(\frac{-b }{ 2 a}) > x_0 $.

• Conditions for both the roots of $f(x)=0$ to be smaller than a specified number ’ $x_{0}$ ’ are $b^2 - 4 a c \geq 0 ;$ $ f(x_0)>0$ & $(\frac{-b }{ 2 a}) < x_0 $.

• Conditions for both roots of $f(x)=0$ to lie on either side of the number ’ $ x_0 $ '

  (in other words the number ’ $ x_0 $ ’ lies between the roots of $ f(x) = 0 $ ), is $ f(x_0)<0$.

• Conditions that both roots of $f(x)=0$ to be confined between the numbers $x_{1}$ and $x_{2},(x_{1}<x_{2})$ are

$\quad$ $\quad$ $\quad$ $\quad$ $b^{2}-4 ac \geq 0 ;$ $f(x_{1})>0 ; $ $f(x_{2})>0 $ & $x_{1} < (\frac{-b }{ 2 a})<x_{2} $.

• Conditions for exactly one root of $f(x)=0$ to lie in the interval $(x_{1}, x_{2})$ i.e. $ x_{1}<x<x_{2}$ is $f(x_{1}) \cdot f(x_{2})<0$.

Roots under special cases

PYQ-2023-Quadratic_Equation-Q1

$\quad$ Consider the quadratic equation $a x^{2}+b x+c=0$

• If $c=0$, then one root is zero. Other root is $-\frac{b}{a}$.

• If $b=0$ The roots are equal but in opposite signs.

• If $b=c=0$, then both roots are zero.

• If $a=c$, then the roots are reciprocal to each other.

• If $a+b+c=0$, then one root is 1 and the second root is $\frac{c}{a}$.

• If $a=b=c=0$, then the equation will become an identity and will satisfy every value of $x$.

Graph of Quadratic equation

$\quad$ The graph of a quadratic equation $a x^{2}+b x+c=0$ is a parabola.

• If $a>0$, then the graph of a quadratic equation will be concave upwards.

• If $a<0$, then the graph of a quadratic equation will be concave downwards.

Maximum and Minimum value

PYQ-2023-Function-Q18

$\quad$ Consider the quadratic expression $a x^{2}+b x+c=0$

• If $a<0$, then the expression has the greatest value at $x=-\frac{b}{2 a}$. The maximum value is $-\frac{D}{4 a}$.

• If $a>0$, then the expression has the least value at $x=-\frac{b}{2 a}$. The minimum value is $-\frac{D}{4 a}$.

Intervals:

$\quad$ Intervals are basically subsets of $\mathrm{R}$ and are commonly used in solving inequalities or in finding domains. If there are two numbers $a, b \in R$ such that $a<b$, we can define four types of intervals as follows :

• Open interval

$$(a, b)={x: a<x<b}$$

• Closed interval

$$[a, b]={x: a \leq x \leq b}$$

• Open-closed interval

$$(a,b] ={x: a<x \leq b} $$

• Closed - open interval

$$ [a, b) ={x: a \leq x<b}$$

• $(a, \infty)={x: x>a}$

• $[a, \infty)={x: x \geq a}$

• $(-\infty, b)={x: x<b}$

• $(\infty, b]={x: x \leq b}$

• $(-\infty, \infty)={x: x \in R}$

Properties of Modulus:

PYQ-2023-Definite_Integration-Q1, PYQ-2023-Definite_Integration-Q5, PYQ-2023-Definite_Integration-Q8, PYQ-2023-MOD-Q3

$\quad$ For any $a, b \in R$

• $|a| \geq 0, \quad|a|=|-a|$

• $|a| \geq a,|a| \geq-a $

• $ |a b|=|a||b| $

• $|\frac{a}{b} \mid=\frac{|a|}{|b|}\quad$

• $|a+b| \leq |a|+|b|$

• $|a-b| \geq|| a|-| b||$