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Complex Numbers and Quadratic Equations is part of NCERT Class 11 Mathematics. It is used for solving equations involving real numbers and imaginary values.
- Complex numbers and quadratic equations help in understanding complex algebra.
- Quadratic equations are a form of polynomial equation which is used to find solutions to the equations.
- It can be represented as:
ax2 + bx + c = 0.
- In this, x denotes the variable, a and b are the numerical coefficients, c is the constant term
- Complex numbers and quadratic equations are expressions of two-dimensional equations in one variable.
- A complex number is a number that can be written in the form of:
x + iy.
- In this, x and y are real numbers, and “i” is called the imaginary number.
- Consider understanding the concept with the help of an example.
- Let x²+3=0 does not have a real solution as the equation becomes x²= -3, which is non-negative.
- In the system of real numbers, D=b² - 4ac <0, so solving such equations is impossible.
- Thus, we make use of complex numbers to solve quadratic equations.
Read More: Complex Numbers & Quadratic Equations Important Questions
Key Terms: Complex Numbers and Quadratic Equations, Complex Number, Quadratic Equations, Real Numbers, Imaginary Number, Solutions, Iota, Roots, Discriminant, Degree
What is a Complex Number?
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Complex Number is any number that is formed as a+ib where a and b are real numbers and i is the imaginary number. It is derived from the nth root of an equation.
- Complex Number is a combination of real and imaginary roots.
- It make use of horizontal axis to represent real part of the equation.
- The vertical axis are used to represent imaginary part of the equation.
- It is used in the field of electromagnetism and control systems.
- The complex in polar form can be represented as
r(cosθ + isinθ)
- where rcosθ is the real part and rsinθ is the imaginary part and i is the value of iota.
Solved Example of Complex NumberExample: 9+3i is a complex numbers. Here i = -1. With this we can say that i² = 1. So, for every equation which does not have a real solution we can use i = -1. |
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What is a Quadratic Equation?
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A quadratic equation is a two dimensional equation that can represented in the form:
ax2 + bx + c = 0.
- Where x denotes the variable.
- a and b are the numerical coefficients
- c is the constant/absolute term.
The equations are used to represen conic section. It is used to define the path of an object.
- Quadratic Equation is derived from a Latin word named quadratus which means square.
- The value of x that satisfies the particular equation is called roots of the equation.
- Each equation provide only two roots which are represented as (α, β).
- The coefficient of the highest degree variable cannot be equal to zero.
- Quadratic equation are used to solve the speed and time problems.
Example of a Quadratic EquationExample 1: Consider throwing a ball in the sky then the trajectory formed by throwing the ball in the sky can be explained with the help of quadratic equations. Example 2: 3x2 – 11x + 23 = 0 is a quadratic equation. |
Read More: R squared formula
Solution of Quadratic Equation
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The solution of a quadratic equation can be found using the formula, x=((-b±√(b²-4ac))/2a). This means it consists of two roots which are (−b + √D / 2a) and (−b - √D / 2a) , where D = b2 – 4ac, called the discriminant of the equation.
- There are three cases that arises for the value of D in the equation which are as follows:
Case 1
In this case the value of D > 0 which results in the formation of two real roots.
Case 2
In this case the value of D = 0 which results in the formation of two equal roots.
Case 3
In this case the value of D < 0 which results in the formation of two imaginary roots.
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Algebra of Complex Numbers
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The algebra of complex numbers are divided into four categories which are as follows:
Addition of two Complex Numbers
Consider z1 = (a+ib) and z2 = (c+id) are two complex numbers. Then addition of two complex numbers can be represented as:
(a + bi) +(c + di) = (a + c) + (b + d)i
Solved Example of Addition of two Complex NumbersExample: z1 = 3+4i and z2 = 4+3i Here a=3, b=4, c=4, d=3 ∴z1+ z2 = (a+c)+(b+d)i ⇒z1 + z2 = (3+4)+(4+3)i ⇒z1 + z2 = 7+7i |
Read More: Pi Formulas
Properties of Addition of two Complex Numbers
The important properties of addition of two complex numbers are as follows:
- Closure law: While adding two complex numbers the resulting number is also a complex number.
- Commutative law: For the complex numbers z1 and z2 , the commutation can be z1+ z2 = z2+z1
- Associative law: While considering three complex numbers, (z1+ z2) + z?3 = z1 + (z2 + z3)
- Additive identity: An additive identity is nothing but zero complex numbers that go as 0+i0. For every complex number z, z+0 = z.
- Additive inverse: Every complex number has an additive inverse denoted as -z.
Difference of two Complex Numbers
Consider z1 = (a + ib) and z2 = (c + id) are two complex numbers. Then difference of two complex numbers can be represented in the same way as the addition of complex numbers, such that
z1 - z2 = z1 + ( -z2)
(a + bi) – (c + di) = (a - c) + (b - d)i
Solved Example of Difference of two Complex NumbersExample: (5+3i) - (2+1i) = (5-2) + (-2-1i) = 3 - 3i |
Multiplication of Complex Numbers
Consider z1 = (a + ib) and z2 = (c + id) are two complex numbers. The product of two complex numbers can be represented as:
z1 * z2 = (ac-bd) + (ad+bc) i
Solved Example of Multiplication of Complex NumbersExample: (5+6i) (2+3i) = (5×2) + (6×3)i = 10+18i |
Properties of Multiplication of complex numbers
The properties of multiplication of complex numbers are similar to the properties we discussed in addition to complex numbers.
- Closure law: When two complex numbers are multiplied the result is also a complex number.
- Commutative law: z1* z2 = z2 * z1
- Associative law: Considering three complex numbers, (z1 z2) z3 = z1 (z2 z3)
- Multiplicative identity: 1+0i is always denoted as 1. This is multiplicative identity. This means that z.1 = z for every complex number z.
- Distributive law: Considering three complex numbers, z1 (z2 + z3) =z1 z2 + z1 z3 and (z1+ z2) z3 = z1 z2 + z2 z3.
Read More: NCERT Solutions for Class 11 Maths Chapter 5: Complex Numbers and Quadratic Equations
Division of two Complex Numbers
Consider z1 = (a + ib) and z2 = (c + id) are two complex numbers. The division of two complex numbers can be represented as:
z1 / z2 = z1 (1/z2 )
(a + ib) / (c + id) = (ac + bd) / (c2 + d2) + i (bc – ad) / (c2 + d2)
Solved Example of Division of two Complex NumbersExample: z1 = 4+2i and z2 = 2 - i z1 / z2 =(4+2i)×1/(2 - i) = (4+i2)(2/(2²+(-1)² ) + i (-1)/(2²+(-1)² )) =(4+i2) ((2+i)/5) = 1/5 [8+4i + 2(-1)+1] = 1/5 [8-2+1+41] = 1/5 [7+4i] |
Read More: Real Numbers Formula
The video below explains this:
Algebraic Operations on Complex Numbers Detailed Video Explanation:
Power of Imaginary Value
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The power of the imaginary value in a complex number are as follows:
- i²= -1
- i³ = i² i=(-1)i = -i
- i4 = (i²)²= (-1)² = 1
- i5 = (i² )² i = (-1)² i = i
- i6 = (i² )³ = (-1)³ = -1
Read More:
| Chapter Related Concepts | ||
|---|---|---|
| Inverse Matrix Formula | Zeroes of Polynomial | Cubic Polynomials |
| Tan2x Formula | Polynomials in One Variable | Transversal |
Square Root of a Real Number
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The square roots of -1 are i,-i. Therefore, (-1) represents i. For real numbers, √a×√b = √ab. If a and b are zero, then √a×√b=√ab=0
- Therefore, (√3i)²= (√3)² i² = 3(-1) = -3
Solved Example of Square Root of a Real NumberExample: Express (5-3i)² in the form of a+ib Answer: (5-3i)² = 5³ -3 × 5² × (3i) + 3 × 5(3i)² - (3i)³ = 125-225i-135+27i = -10-198i |
Read More: Exponential Growth Formula with Solved Examples
Argand Plane and Polar representation
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When a plane has a complex number that is allocated to each point then it is called a complex plane or the argand plane. In this plane, the modulus of a complex number is defined as the distance between the given point and the origin.
- The points on the x-axis are complex numbers of the form a+i0
- The points on the y-axis are complex number of the form 0+ib
- They are called real axis and imaginary axis respectively.
The above diagram is a representation of the complex number and its conjugate. Here, complex number z=x+iy is denoted as P(x, y) and its conjugate z=x-iy is denoted as Q(x, -y)
Polar representation
The polar form of a complex number is given as r=√(x²+y² )=|z| is the modulus value of z and θ. There are many intervals to be considered in polar forms.
Example of Polar RepresentationExample: For a complex number if z≠0, then there's only one value of θ in 0≤θ≤2π. Similarly, there are other such intervals like -π<θ≤π. This is called the principal argument of z and is denoted as argz. |
Read More: Inverse Trigonometric Formulas
The video below explains this:
Quadratic Equations Detailed Video Explanation:
Euler’s formula
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The Euler’s formula for any real number y is given by
eiy=cosy+isiny
- If x and y are real numbers, then,
e(x+iy)=ex (cosy+isiny)=ex cosy+iex siny
- This is called the decomposition of the exponential function into its real and imaginary parts.
- Euler’s formula is used to explain the rotation along the unit circle in the complex plane.
- It is used for solving de Moivre’s theorem.
Solved Example of Euler’s FormulaExample: Find the value of e iπ/3. Ans. Given eiπ/3 Using Euler’s formula: eix = cos x + i sin x
|
Read More: Algebra Formulas for Class 10
Things to Remember
- Complex numbers and quadratic equations are used to solve for unknown values of roots.
- While solving a quadratic equation, each complex solution will have a conjugate partner.
- In electronics, complex numbers are used to represent voltage and current equations.
- The trajectory pathway of a cliff jumper can be explained using quadratic equations.
- Descartes's theorem is an application of quadratic equations.
Read More: Angle of Depression
Sample Questions
Ques.Where does z lies, if | z – 5i / z + 5i | = 1. (3 marks)
Ans. The solution is as follows:
Ques. If (a2+1)2 / 2a –i = x + iy, then find the value of x2 + y2. (3 marks)
Ans. The solution is as follows:
Ques. Find the conjugate of 2-i / (1 – 2i)2. (2 marks)
Ans. The solution is as follows:
Ques. Solve the equations Re (z2) = 0, |z| = 2. (3 marks)
Ans. The solution is as follows:
Ques. If for complex number z1 and z2, arg (z1) – arg (z2) = 0, then show that |z1 – z2| = | z1|- |z2 |. (4 marks)
Ans. The solution is as follows:
Ques. If Z1, Z2 and Z3, Z4 are two pairs of conjugate complex numbers, then find arg (Z1/ Z4) + arg (Z2/ Z3). (2 marks)
Ans. The solution is as follows:
Ques. If |z1 | = 1 (z1≠ -1) and z2 = z1 – 1/ z1 + 1 , then show that the real part of z2 is zero. (4 marks)
Ans. The solution is as follows:
Ques. Express (7-3i)² in the form of a+ib. (2 marks)
Ans. (7-3i)² = 7³ -3 × 7² × (3i) + 3 × 7(3i)² - (3i)³
= 343-441i-567+27i
= -224-414i
Ques. Suppose z1 = 8 +4i and z2 = 9+3i then add these two complex numbers. (2 marks)
Ans. Here a=8, b=4, c=9, d=3
∴z1+ z2 = (8+9)+(4+3)i
⇒z1 + z2 = 17+7i
Ques. Find out the roots of the quadratic equation x2 - 3x - 9 = 0 using the quadratic formula. (2 marks)
Ans. In the given quadratic equation,
a = 1, b = -3, and c = -9
Using the Quadratic Formula,
x = [-b ± √(b2 - 4ac)]/2a
x = [-(-3) ± √((-3)2 - 4(1)(-9))]/2(1)
x = [3 ± √36] / 2
x = [3 ± 6] / 2
x = (3 + 6)/2 or (3 - 6)/2
x = 9/2 or -3/2
Ques. Suppose z1 = 6 +4i and z2 = 9+2i then subtract and multiply these two complex numbers. (3 marks)
Ans. Here a=6, b=4, c=9, d=2
∴z1- z2 = (6-9)+(4-2)i
⇒z1 - z2 = -3+2i
Multiplication of two complex number is given as:
Here a=6, b=4, c=9, d=2
∴z1 . z2 = (6x9 – 4x2) +(6x2 + 4x9)i
⇒z1 . z2 = 46+48i
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