[Maths Class Notes] on Complex Numbers and Quadratic Equations Pdf for Exam

A Guide to the Complex Numbers and Quadratic Equations

Mathematics includes a lot of topics that give an edge to your problem-solving abilities and critical thinking. Complex numbers and quadratic equations is a segment of maths that deals with crucial theorems and concepts along with various formulae. It comprises of linear and quadratic equations along with roots related to the complex number’s set (known as complex roots).

Although maths is a scoring subject, yet we find problems tricky because of insufficient knowledge of different topics. If you get stuck with cumbersome mathematical problems, try seeking complex numbers and quadratic equations NCERT solutions. 

Let us take a tour for a better understanding.

Define Complex Numbers

A mathematical equation having a complex number comprises of the real and imaginary sections. Complex numbers are nothing but a combination of two numbers (real, imaginary). Real ones mostly comprise of 1, 1998, 12.38, whereas imaginary numbers generate a negative result when they get squared.

For instance, consider an equation in the form (a + ib). Here, both a and b constitutes a complex number having a, as the real portion of the complex number and b acts as an imaginary one.

What are Quadratic Equations?

A quadratic equation is a mathematical equation in algebra that comprises of squares of a variable. It derives the name from the word ‘quad’ which implies square. It is also known as ‘equation of a degree 2’ (because of x2).

In complex numbers and quadratic equations, the standard form of a quadratic equation appears as:

  ax2 + bx + c = 0

Where x is a variable or an unknown factor, and a, b and c are known values.

The below chart show a few examples of a quadratic equation: 

Exercises on Complex Nos. and Quadratic Equations

Below there is a complex numbers and quadratic equations miscellaneous exercise. Go through it carefully!

1. Will be the Equation of the Following if they have Real Coefficients with One Root? 

1. 1 -2i

2. -2 – i√3

3. 1/(2 + i√2)

Solution: Assume, (a + b) and (a – b) are roots for all the problems.

1. Sum of the value of roots is (1 + 2i) + (1 – 2i) = 2 

Then, products of these roots will be (1 + 2i) * (1 – 2i) = 1 + 4 = 5

Therefore, the equation is equal to x2 – 2x + 5 = 0

2. Here, sum of the value of roots is -2 – i√3 – 2 + i√3 = – 4

Multiplication of these roots is (-2 – i√3) * (- 2 + i√3) = – 3 – 4 = 7

Hence, the quadratic equation will be x2 + 4x – 7 = 0

3. Sum of the roots will be (2 + i√2)/2 + (2 + i√2)/2 = 2 (If, 1/(2 + i√2) = (2 + i√2)/2)

Multiplication of them will result in (4 + 2)/4 = 3/2

Hence, the equation is equal to 2x2 – 4x + 3 = 0

Rack Your Brains

Read the following question on complex numbers and quadratic equations thoroughly to excel in your studies.

1. Suppose z = (√3/2 + i/2)5 + (√3/2 – i/2)5. If the real portion is R(z) and the imaginary section constitutes I(z) of the value z, then what will be the answer?

1. R(z) > 0 and I(z) > 0

2. R(z) < 0 and I(z) > 0

3. I(z) = 0 and R(z) < 0

4. R(z) = 3

5. I(z) = 0

6. None of them

Some chapters apart from quadratic equations can appear problematic to solve, such as integrals, permutation and combination, etc. However, you find solutions to these problems quickly if you understand the theorems well. Also, you can try searching for complex numbers and quadratic equations solutions to tackle cumbersome topics.

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