ax

A real number α is called the root or solution of the quadratic equation if

aα

we can learn in details from below links

- Quadratic Polynomial
- Graphing quadratics Polynomial
- what is a quadratic equation
- How to Solve Quadratic equations
- Factoring quadratics equations
- Solving quadratic equations by completing the square
- Solving quadratic equations by using Quadratic formula
- Nature of roots of Quadratic equation
- Problem based on discriminant of a quadratic equation
- Quadratic word problems

- The root of the quadratic equation is the zeroes of the polynomial p(x).
- We know from chapter two that a polynomial of degree can have max two zeroes. So a quadratic equation can have maximum two roots
- A quadratic equation has no real roots if b
^{2}- 4ac < 0

x

Solution

There is no constant term in this quadratic equation, we can x as common factor

x(x-6)=0

So roots are x=0 and x=6

2. Solve the quadratic equation

x

Solution

x

x

or x=4 or -4

3. Solve the quadratic equation by factorization method

x

Solution

1) First we need to multiple the coefficient a and c.In this case =1X-20=-20

The possible multiple are 4,5 ,2,10

2) The multiple 4,5 suite the equation

x

x(x-5)+4(x-5)=0

(x+4)(x-5)=0

or x=-4 or 5

4. Solve the quadratic equation by Quadratic method

x

Solution

For quadratic equation

ax

roots are given by

$x=\frac{-b+\sqrt{b^{2}-4ac}}{2a}$

and

$x=\frac{-b-\sqrt{b^{2}-4ac}}{2a}$

Here a=1

b=-3

c=-18

Substituting these values,we get

x=6 and -3

We have already studied about Complex number in previous chapter,Now it times to use in quadratic equation

We know that if

b

We dont have real roots

Now if b

So now we can define imaginary roots of the equation as

$k_{1}=\frac{-b+i\sqrt{4ac-b^{2}}}{2a}$

$k_{2}=\frac{-b-i\sqrt{4ac-b^{2}}}{2a}$

The quadratic equations containing complex roots can be solved using the Factorization method,square method and quadratic method explained above

So far we have read about quadratic equation where the coefficent are real. Complex quadratic equation are the equations where the coefficent are complex numbers.

These quadratic equation can also be solved using Factorization method,square method and quadratic method explained above

The roots may not conjugate pair in these quadratic equations

(1) Solve the quadratic equation

x

Solution

x

(x-1)

(x-1-3i)(x-1+3i)=0

So roots are

(1+3i) and (1-3i)

We can also obtain the same things using quadratic methods

$k_{1}=\frac{-b+i\sqrt{4ac-b^{2}}}{2a}$

$k_{2}=\frac{-b-i\sqrt{4ac-b^{2}}}{2a}$

$k_{1}=\frac{2+i\sqrt{40-2^{2}}}{2}$=1+3i

$k_{1}=\frac{2-i\sqrt{40-2^{2}}}{2}$=1-3i

(2) For which values of k does the polynomial x

Solution:

For the roots to be complex conjugate ,we should have

b

16-4k <0

or k> 4

**Notes**- What is complex numbers
- Algebra Of complex Numbers
- Conjugate of Complex Numbers
- Modulus of complex numbers
- Argand Plane
- Polar Representation of the complex number
- Rotation of Complex Number
- Identities for Complex Numbers
- Eulers formula and De moivre's theorem
- Cube Root of unity
- Complex roots of Quadratic equations

**NCERT Solutions & Worksheets**