While solving the equation of the form \(ax^{2}+bx+c = 0\), the roots of the equations can take three forms which are as follows:
Two Distinct Real roots
Similar Root
No Real roots (Complex Roots)
The introduction of complex numbers in the 16th century made it possible to solve the equation \(x^{2}+1 = 0\). The roots of the equation are of the form \(x = \pm \sqrt{-1}\) and no real roots exist. Thus, with the introduction of complex numbers, we have Imaginary roots.
We denote \(\sqrt{-1}\) with the symbol i, where i denotes Iota (Imaginary number).
An equation of the form z= a+ib, where a and b are real numbers, is defined to be a complex number. The real part is denoted by Re z = a and the imaginary part is denoted by Im z = b.
Algebraic Operation on Complex numbers:
- Addition of two complex numbers
- Subtraction of two complex number
- Multiplication of two complex number
- Division of two complex number
Power of Iota(i):
Depending upon the power of i, it can take the following values
\(i^{4k+1} = i\)
\(i^{4k+2} = -1\)
\(i^{4k+3} = -i\)
\(i^{4k} = 1\)
where k can have any integral value (positive or negative).
Similarly we can find for the negative power of i, which are as follows
\(i^{-1}=\frac{1}{i}\)
Multiplying and dividing the above term with i, we have
\(i^{-1}=\frac{1}{i} \times \frac{i}{i}\)
\(i^{-1}=\frac{i}{i^{2}} = \frac{i}{-1} = -i\)
Note: \(\sqrt{-1}\times \sqrt{-1} = \sqrt{-1 \times -1} = \sqrt{1} = 1 \) contradicts to the fact that \(i^{2}= -1\).
Therefore for an imaginary number, \(\sqrt{a}\times \sqrt{b} \neq \sqrt{ab}\).
Identities:
(i) \(\left ( z_{1}+z_{2} \right )^{2} =z_{1}^{2}+z_{2}^{2}+2z_{1}z_{2}\)
(ii) \(\left ( z_{1}-z_{2} \right )^{2} =z_{1}^{2}+z_{2}^{2}-2z_{1}z_{2}\)
(iii) \(z_{1}^{2}-z_{2}^{2} = \left ( z_{1} + z_{2} \right )\left ( z_{1} – z_{2} \right )\)
(iv) \(\left ( z_{1}+z_{2} \right )^{3} = z_{1}^{3}+3z_{1}^{2}z_{2}+3z_{1}z_{2}^{2}+z_{2}^{3}\)
(v) \(\left ( z_{1}-z_{2} \right )^{3} = z_{1}^{3}-3z_{1}^{2}z_{2}+3z_{1}z_{2}^{2}-z_{2}^{3}\)
Modulus and Conjugate of a Complex number:
Let z=a+ib be a complex number.
The Modulus of z is represented by \(\left | z \right |\).
Mathematically, \(\left | z \right |= \sqrt{a^{2}+b^{2}}\)
The conjugate of z is denoted by \(\bar{z}\).
Mathematically, \(\bar{z}= a – ib\)
Let’s Work Out Example: Express the following in a+ib form. \(\frac{5+ \sqrt{3}i}{1-\sqrt{3}i}\). And then find the Modulus and Conjugate of the complex number.] Solution: Given \(\frac{5+ \sqrt{3}i}{1-\sqrt{3}i}\) z= \(\frac{5+ \sqrt{3}i}{1-\sqrt{3}i} \times \frac{1+\sqrt{3}i}{1+\sqrt{3}i} = \frac{5-3+6\sqrt{3}i}{1+3}=\frac{1}{2} + \frac{3}{2}i\)
Modulus, \(\bar{z} = \sqrt{\left ( \frac{1}{2}\right )^{2} + \left (\frac{3}{2}\right )^{2}}\) \(\bar{z} = \sqrt{\left ( \frac{10}{4}\right )}\) \(\bar{z} = \frac{\sqrt{10}}{2}\)
Conjugate, \(\bar{z} = \frac{1}{2} – \frac{3}{2}i\) |
Argand Plane & Polar Representation of Complex Number:
Similar to the XY plane, the Argand(or complex) plane is a system of rectangular coordinates in which the complex number a+ib is represented by the point whose coordinates are a and b.
We find the real and complex components in terms of r and θ, where r is the length of the vector and θ is the angle made with the real axis.
Quadratic Equation:
The Quadratic equation is of the form \(ax^{2}+bx+c=0\)
where the roots of the equation are given as
\(\large x= \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}\)
or, \(\large x= \frac{-b \pm \sqrt{D}}{2a}\)
where Discriminant (D)\( = b^{2}-4ac\)
For an equation having complex roots, D<0
Let’s Work Out- Example- Find the roots of the given quadratic equation \(27x^{2}-10x+1=0\) Solution- Given \(27x^{2}-10x+1=0\) The roots of the equation is given as \(\large x= \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}\) \(\large x= \frac{-(-10)\pm \sqrt{(-10)^{2}-4\times 27 \times 1}}{2 \times 27}\) \(\large x= \frac{10\pm \sqrt{-8}}{54}\) \(\large x= \frac{5}{27} \pm \frac{\sqrt{2}i}{27}\) |
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