Complex Numbers and Quadratic Equations - Topics, Books, Preparation Tips FAQs

Can there be a negative number inside the square root? Consider the function $f(x) = x^2 +1$. There is no solution for this function in real number system. The solution for this function is $x= \sqrt{-1}$. When we square a real number it is impossible to get a negative real number. For having solution for this equation, we must create an imaginary number as a square root of $−1$. So, this imaginary part along with the real part is called a complex number.

Complex numbers are useful in representing a phenomenon that has two parts varying at the same time, for instance an alternating current. Engineers, doctors, scientists, vehicle designers and others who use electromagnetic signals need to use complex numbers for strong signal to reach its destination. Quadratic equations are polynomial equations with degree $2$. This article is about the concept of complex numbers and quadratic equations class 11. This article includes complex numbers and quadratic equations notes, complex numbers and quadratic equations class 11 formulas, list of topics included in JEE Main mathematics complex numbers and quadratic equations, important books for complex numbers and quadratic equations and tips for preparing for the topic complex numbers and quadratic equations.

Complex Numbers

The number which can be expressed in the form $x+iy$ is called a complex number. Here $x$ and $y$ are real numbers and $i$ is the imaginary part called "iota".

For example, $z=5+2 i$ is a complex number.

Here $5$ is called the real part and is denoted by $\operatorname{Re}(\mathrm{z})$, and $2$ is called the imaginary part and is denoted by $\operatorname{Im}(z)$

Iota is an imaginary unit number that is denoted by $i$ and the value of iota is $\sqrt{-1} \text { i.e., } i=\sqrt{ -1}$. For analyzing the powers of iota, we calculate the first few powers of iota. i.e $i=\sqrt{-1}$, $i^2=-1$, $i^3=-\sqrt{-1}$, $i^4=1$

If the imaginary part of the complex number is zero, then the complex number becomes a real number. Eg. $2+0i$ is a real number as $2+0i = 2$.

If the real part of the complex number is zero, then the complex number becomes a purely imaginary number. Eg. $0+2i = 2i$

Quadratic Equations

The highest power of the variable in the polynomial expression is called the degree of the polynomial.

A polynomial equation in which the highest degree of a variable term is $2$ is called a quadratic equation.
Standard form of quadratic equation is $a x^2+b x+c=0$
Where $\mathrm{a}, \mathrm{b}$, and c are constants (they may be real or imaginary) and called the coefficients of the equation and $a \neq 0$ (a is also called the leading coefficient).

$
E g,-5 x^2-3 x+2=0, x^2=0,(1+i) x^2-3 x+2 i=0
$

As the degree of the quadratic polynomial is $2$, so it always has $2$ roots (number of real roots + number of imaginary roots $=2$ )

Roots of Quadratic Equations

If $f(x)$ is a polynomial, then $f(x)=0$ is called a polynomial equation.
The value of $x$ for which the polynomial equation, $f(x)=0$ is satisfied is called a root of the polynomial equation.
If $x=\alpha$ is a root of the equation $f(x)=0$, then $f(\alpha)=0$.
$\mathrm{Eg}, \mathrm{x}=2$ is a root of $\mathrm{x}^2-3 x+2=0$, as $x=2$ satisfies this equation.
A polynomial equation of degree $n$ has $n$ roots (real or imaginary). So, the quadratic equation has $2$ roots as the degree is $2$.

Complex Numbers and Quadratic Equations Formulas

Complex Numbers Formula

Conjugate of Complex Numbers

The Conjugate of a complex number $z$ is represented by $\bar{z}$. while ($z$ & $\bar{z})$ together are known as a complex-conjugate pair because $z$ and $\bar{z}$ are conjugate to each other.

The conjugate of a complex number $\mathrm{z}=\mathrm{a}+\mathrm{ib}$ ( $\mathrm{a}, \mathrm{b}$ are real numbers) is $\mathrm{a}-\mathrm{ib}$. It is denoted as $\bar{z}$. if $z=a+i b$, then its conjugate is $z=a-i b$.

The conjugate of complex numbers is obtained by changing the sign of the imaginary part of the complex number. The real part of the number is left unchanged.

Modulus of Complex Numbers

If $z=x+i y$ is a complex number. Then, the modulus of complex number $z$, denoted by $\mid z$ l, is the distance of $z$ from the origin in the Argand plane, and it is a non-negative real number equal to $\sqrt{x^2+y^2}$.
i.e. $|z|=\sqrt{x^2+y^2}$.

Every complex number can be represented as a point in the argand plane with the $x$-axis as the real axis and the $y$-axis as the imaginary axis.
$|z|=\sqrt{x^2+y^2}=r$ (length r from origin to point $\left.(\mathrm{x}, \mathrm{y})\right)$

Square Root of Complex Numbers

A complex number's square root is equal to another complex number whose square equals the original complex number. For example, if $\sqrt{ }(a+i b)=x+i y$ is the square root of the complex number $a+i b$, then $(x+i y) 2=a+i b$. Finding the values of x and y by squaring both sides of the equation $\sqrt{ }(a+i b)=x+i y$ and comparing the real and imaginary parts is one easy method to obtain the square root of a complex integer, $a + ib$. Let's look at the formula for calculating a complex number's square root.

$
\sqrt{\mathrm{z}}= \pm\left(\sqrt{\frac{|z|+\operatorname{Re}(z)}{2}}+\mathrm{i} \sqrt{\frac{|z|-\operatorname{Re}(z)}{2}}\right)
$

if $\operatorname{Im}(z)>0$ otherwise there will be a $-ve$ sign between the real and imaginary parts of the square root of $z$.

Algebraic Operations on Complex Numbers

The algebraic operations on complex numbers include addition, subtraction, multiplication and division.

Addition of Complex Numbers: Let us say that $z 1$ and $z 2$ are the two complex numbers

$
\begin{aligned}
& z_1=(a+i b) \\
& z_2=(c+i d)
\end{aligned}
$
On addition, these two complex numbers will be equal to

$
\begin{aligned}
& z_1+z_2=(a+i b)+(c+i d) \\
& =(a+c)+i(b+d)
\end{aligned}
$

Subtraction of Complex Numbers: Let us say that $z1$ and $z2$ are two complex numbers.

$
\begin{aligned}
& z_1=(a-i b) \\
& z_2=(c-i d)
\end{aligned}
$
On deduction of these two complex numbers will be equal to

$
\begin{aligned}
& z_1-z_2=(a-i b)-(c-i d) \\
& =(a-c)-i(b-d)
\end{aligned}
$

Which gives another complex number whose real part of the new complex number is $\operatorname{Re}(z 1)-\operatorname{Re}(z 2)=[a-c]$ and the imaginary part of the new complex number $=\operatorname{Im}(Z 1)-\operatorname{Im}(Z 2)=[b-d]$.

Examples:-
- $(6+10 i)-(10+3 i)=(6-10)+(10-3) i=-4+7 i$
- $(2+3 i)-(2+2 i)=(2-2)+(3-2) i=-0+1 i$

Which gives another complex number whose real part is $\operatorname{Re}(z 1)+\operatorname{Re}(z 2)=a+c$ and the imaginary part of the new complex number $=\operatorname{Im}(z 1)+\operatorname{Im}(z 2)=b+d$

Example: $(2+3 i)+(2+2 i)=(2+2)+(3+2) i=4+5 i$

Multiplication of Complex Numbers: Multiplying any two complex numbers is equal to the multiplication of two binomials.

Let us assume that $Z1= a + ib$ and $z2= c + id$.

On multiplying we obtain:-

$\lgroup\:a + bi\rgroup \lgroup\:c + di\rgroup = \lgroup\:a + bi\rgroup c + \lgroup\:a + bi\rgroup di$

$\lgroup\:a + bi\rgroup\lgroup\:c + di\rgroup = \lgroup\:a c +\lgroup\:b c\:\rgroup\:i\:\rgroup+\lgroup\:\lgroup\:a d\rgroup\:i + b d −1\rgroup$

$\lgroup\:a + bi\rgroup\lgroup\:c + di\rgroup = \lgroup\:ac − b d + \lgroup\:[b c + a d\rgroup\rgroup$

Examples:-

$2i \lgroup2 + 6i\rgroup$

can be viewed as $\lgroup0 + 2i\rgroup \lgroup2 + 6i\rgroup$

$= 2i \lgroup2 + 6i\rgroup$

$= 2i + 18i^{2}$

$=2i-18$

$=-18+2i$

Division of Complex Numbers: Let us assume any complex number $z_1=c_1+id_1$ and $z_2=c_2+id_2$ then the quotient of $\frac{z_1}{z_2}$ is equal to,

$\frac{z_1}{z_2}=z_1. \frac{1}{z_2}$

Thus, to find $\frac{z_1}{z_2}$

we have to multiply $z1$ with the inverse of $z2$.

Quadratic Equations Formula

How to Solve Quadratic Equations?

Solving quadratic equations means finding a value (or) values of variable which satisfy the equation. The value(s) that satisfy the equation is known as its roots (or) solutions (or) zeros. Since the degree of the quadratic equation is $2$, it can have a maximum of $2$ roots. For example, one can easily see that $x=1$ and $x=2$ satisfy the quadratic equation $x^2-3 x+2=0$ (you can substitute each of the values in this equation and verify). Thus, $x=1$ and $x=2$ are the roots of $x^2-3 x+2=0$. But how to find them if they are not given?

There are different ways of solving quadratic equations.

• Solving quadratic equations by factoring
• Solving quadratic equations by completing the square
• Solving quadratic equations by graphing
• Solving quadratic equations by quadratic formula

Nature of Roots

The nature of roots of quadratic equations are,

1. Real and Distinct Roots

• The discriminant is positive, that is, $b^2-4 a c>0$.
• The curve intersect the $x$-axis at two distinct points.

2. Real and Equal Roots

• The discriminant is equal to zero, that is, $b^2-4 a c=0$.
• The curve intersects the $x$-axis at only one point.

3. Complex Roots

• The discriminant is negative, that is, $b^2-4 a c<0$.
• The curve does not intersect the $x$-axis.

How to find the nature of the roots?

Step 1: Compare the given quadratic equation with the standard form of quadratic equations $a x^2+b x+c=0$ and find the values for the coefficients $a, b$, and $c$.

Step 2: Substitute the value of the coefficients in the discriminant equation $b^2-4 a c$ and solve it

Step 3: Observe the value you get for the discriminant. If it is less than zero you have complex roots. If it is equal to zero you have real and equal roots. If it is greater than zero you have real and distinct roots.

Assume $\alpha$ and $\beta$ to be the roots of a quadratic equation.

Sum of roots: $
\begin{aligned}
& a+\beta=(-b+\sqrt{ } D) / 2 a+(-b-\sqrt{ } D) / 2 a \\
& =(-b / 2 a)+(\sqrt{ } D / 2 a)-(b / 2 a)-(\sqrt{ } / 2 a) \\
& =-2 b / 2 a \\
& =-b / a
\end{aligned}
$

Product of roots: $
\begin{aligned}
& a . \beta=(-b+\sqrt{ } D) / 2 a .(-b-\sqrt{ } D) / 2 a \\
& =\left\{(-b)^2-(\sqrt{ } D)^2\right\} /(2 a)^2 \\
& =\left(b^2-D\right) / 4 a^2 \\
& =\left(b^2-b^2+4 a c\right) / 4 a^2\left[D=b^2-4 a c\right] \\
& =c / a
\end{aligned}
$

A quadratic equation can be formed using the sum and product of the roots.

$
x^2-[a+\beta] x+[\alpha \cdot \beta]=0
$

Location of Roots

Now let us look into the location of roots of quadratic equations. Let $\mathrm{f}(\mathrm{x})=\mathrm{ax}{ }^2+\mathrm{bx}+\mathrm{c}$, where $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are real numbers and ' a ' is non-zero number. Let $x_1$ and $x_2$ be the roots of the equation, and let $k$ be a real number. Then:

1. If both roots of $f(x)$ are less than $k$ then

i) $\mathrm{D} \geq 0$ (as the real roots may be distinct or equal)

ii) $af( k)>0$ (In both the cases $af(\mathrm{k})$ is positive, as in second case if a $<0$ then $\mathrm{f}(\mathrm{k})<0$, so multiplying two $-ve$ values will give us a positive value)

iii) $k>\frac{-b}{2 a}$ since $\frac{-b}{a}$ will lie between $x_1$ and $x_2$, and $x_1$, $x_2$ are less than k so $\frac{-\mathrm{b}}{2 \mathrm{a}}$ will be less than k .

2. If both roots of $f(x)$ are greater than $k$

i) $\mathrm{D} \geq 0$ (as the real roots may be distinct or equal)

ii) $\mathrm{af}(\mathrm{k})>0$ (In both the cases af( k$)$ is positive, as in second case if a $<0$ then $\mathrm{f}(\mathrm{k})<0$, so multiplying two -ve values will give us a positive value)

iii) $k<\frac{-\mathrm{b}}{2 \mathrm{a}}$ since $\frac{-\mathrm{b}}{2 \mathrm{a}}$ will lie between $x_1$ and $x_2$, and $x_1$, $x_2$ are greater than $k$ so $\frac{-\mathrm{b}}{2 \mathrm{a}}$ will be greater than $k$.

Condition for number $k$: Let $\mathrm{f}(\mathrm{x})=\mathrm{ax}^2+\mathrm{bx}+\mathrm{c}$ where $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are real numbers and ' a ' is non-zero number. Let $x_1$ and $x_2$ be the real roots of the function. And let $k$ is any real number. Then:

1. If lies between the root $x_1$ and $x_2$

i) $\mathrm{D} \geq 0$ (as the real roots may be distinct or equal)

ii) $k_1<\frac{-b}{2 a}<k_2$, where $a \leq 3$ and $k_1<k_2$

2. If $k_1, k_2$ lies between the roots

af $\left(\mathrm{k}_1\right)<0$ and af $\left(\mathrm{k}_2\right)<0$

List of Topics According to NCERT/JEE MAIN

Applications of Complex Numbers and Quadratic Equations

Complex numbers and quadratic equations both find wide range of application in real-life problem, for example in physics when we deal with circuit and if circuit is involved with capacitor and inductance then we use complex numbers to find the impedance of the circuit and for doing so we use complex numbers to represent the quantities of capacitor and inductance responsible in contribution of impedance. Complex numbers have essential concrete applications in signal processing, control theory, electromagnetism, fluid dynamics, quantum mechanics, cartography, and vibration analysis.

Quadratic equations are basically parabolic so they are used in throwing a ball (studying the trajectory of ball as equation of trajectory is parabolic), hitting a golf ball, the angle at which we hit golf ball decide the trajectory(parabolic path) of golf ball and hence the distance covered, in rocket propulsion also quadratic equations are used in the calculation of the trajectory of propulsion.

Importance of Class 11 Complex Numbers and Quadratic Equations

Complex Numbers and Quadratic Equations have a significant weighting in the IIT JEE test, which is a national level exam for 12th grade students that aids in admission to the country's top engineering universities. It is one of the most difficult exams in the country, and it has a significant impact on students' futures. Several students begin studying as early as Class 11 in order to pass this test. When it comes to math, the significance of these chapters cannot be overstated due to their great weightage. You may begin and continue your studies with the standard books and these revision notes, which will ensure that you do not miss any crucial ideas and can be used to revise before any test or actual examination.

NCERT Notes Subject Wise Link:

• NCERT notes Class 11 Maths

• NCERT notes Class 11 Physics

• NCERT notes Class 11 Chemistry

• NCERT notes Class 11 Biology

How to Study Complex Numbers and Quadratic Equations?

Start with understanding basic concepts like Definition of Complex Number, Integral Powers of iota (), Representation of a Complex number in various forms. Then go to the algebra of complex numbers, Argand plane, modulus and argument of complex number. After studying these concepts go through solved examples and then go to mcq and practice the problem to make sure you understood the topic. Solve the questions of the books which you are following and then go to previous year papers. For the quadratic equation, you should do the same things in the same order as mentioned above for complex numbers. While going through concept make sure you understand the derivation of formulas and try to derive them by your own, as many times you will not need the exact formula but some steps of derivation will be very helpful to solve the problem if you understand the derivation it will boost your speed in problem-solving. Since both topics are geometrically involved try to understand and relate the things with geometry and graph as the graph makes visualization of the problem easy and hence it makes the problem easy for us to solve. At the end of chapter try to make your own short notes for quick revision, make a list of formula to revise quickly before exams or anytime when you required to revise the chapter, it will save lots of time for you.

If you are preparing for competitive exams then solve as many problems as you can. Do not jump on the solution right away. Remember if your basics are clear you should be able to solve any question on this topic.

Important Books for Complex Numbers and Quadratic Equations

Start from NCERT Books, the illustration is simple and lucid. You should be able to understand most of the things. Solve all problems (including miscellaneous problem) of NCERT. If you do this, your basic level of preparation will be completed.

Then you can refer to the book Algebra Arihant by Dr. SK goyal or RD Sharma or Cengage Mathematics Algebra. Complex Numbers and Quadratic Equations are explained very well in these books and there are an ample amount of questions with crystal clear concepts. Choice of reference book depends on person to person, find the book that best suits you the best, depending on how well you are clear with the concepts and the difficulty of the questions you require.

NCERT Solutions Subject wise link:

• NCERT solutions for class 11 Physics.

• NCERT solutions for class 11 Chemistry.

• NCERT solutions for class 11 Mathematics.

• NCERT solutions for class 11 Biology.