Complex Numbers and Quadratic Equations

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: 

Equations

Detailed Explanations

3x2 + 4x + 6 = 0

In this expression, the known values a = 3, b = 4 and c = 6; while x remains the unknown factor.

2x2 – 6x = 0


Here, the known factors a = 2 and b = 6. However, can you ascertain the value of c? Well, the value of c = 0 as it is not present.

7x – 4 = 0

Here the value of a is equal to zero since the equation is not quadratic.


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

  1. 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

  1. 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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FAQ (Frequently Asked Questions)

1. How Can You Add Two Complex Numbers?

Ans. In mathematics, complex numbers can be obtained when a real number combines with an imaginary one. In a complex equation, there are two numbers present; one constituting a real number (like 1, √2, ¾, 1998, etc.) and the other being imaginary one (3i, 3i/4, (√4)i and 1.09i, etc.).

It is easy to add two complex numbers. Add the real numbers and imaginary ones separately.

For example, suppose there are two numbers (a + ib) and (c + id), then the steps will be:

= (a + ib) and (c + id),

= (a + c) + (b + d)i

2. Which Quadratic Equation Contains Imaginary Roots?

Ans. A quadratic equation comprises of imaginary roots when an equation’s discriminant is negative (the segment lying under the square root sign, such as b2 – 4ac). Since we know, imaginary numbers always generate a negative value when squared. When the discriminant value is negative, the answers do not appear real. 

When you use quadratic equations, it often generates two values. When the number under the square root appears negative, the answer is known as complex conjugates. The numbers will be (r + si) and (r – si), and these numbers have both real and imaginary values. Here r is the real value, and si has the imaginary segments.

3. What will be the Answer of the Equation tan [ilog (a+ib/a−ib)] by Using the Quadratic Formula?

Ans. Quadratic equations come from the ‘quad’ which means square. Moreover, quadratum is the Latin word for square. It is a type of problem that indulges a variable as it multiples by itself. This process occurs through squaring of a number (such as x multiplied by itself resembles quadratic equation in an expression). 

It is also popularly known as ‘equation of a degree 2’ because the characteristics of squaring. Suppose, we need to find the value of tan [ilog (a+ib/a−ib)] 

 a – ib = √a2 + √b2e-iΘ

    Therefore, (ilog (√a2 + √b2e-iΘ/√a2 + √b2e))

     Tan (i loge-2iΘ) = tan (2Θ) 

     = 2 tan Θ/ (1 – tan2 Θ) = 2ab/ (a2 – b2)

4. How can one Find Solution of a Quadratic Equation through Factoring?

Ans. A quadratic equation is an expression where a square of a variable is present (such as x2). It includes both known and unknown values in the expression. You can solve a quadratic equation through factoring method, following the method detailed out below:

First, you need to keep all the terms on the side of the equal sign in a way that only zero is left on the other side. Then, select a factor; the factor must be zero and solve the equation in that way. You can also check the answer by inserting it in the original quadratic expression.