Roots of Quadratic Equation Examples and Formula

What is Quadratic equation with Formula?

A quadratic equation in x is an equation that can be written in the form of ax² + bx + c = 0; where a,b and c are. real numbers and a≠0.

Another name for a quadratic equation in x is 2nd degree polynomial in x.


Quadratic equation








Roots of Quadratic Equation Examples

(i) x² - 7x + 10 = 0 ; a = 1 , b = -7 , c = 10

(ii) 6x² + 4x - 13 = 0 ; a=6 , b = 4 , c = -13

(iii) x² = 8 ; a = 1 , b = 0 , c = -8  

(iv) 3x² - x = 0 ; a = 3 , b = -1 , c = 0


   Solutions of Quadratic equation

There are three basic techniques for solving a quadratic equation:

  •  By factorization
  •  By completing square , extracting square root
  •  By applying the quadratic formula


factorization:

It involves factoring the polynomial ax² + bx + c. It makes use of the facts that if ab=p , then a = 0 or b=0

Examples of factorization

x² -7x+10 = 0

(x-2) (x-5) = 0

either       x-2 = 0 ⇒ x = 2

Or            x-5 = 0  ⇒x = 5

so, solution set = {2,5} 

The solution of an equation are also called its roots.

in the upper example 2 and 5 are roots of the equation x² -7x+10 = 0

By completing square:

Sometimes, quadratic polynomials are not easily factorable.

for example, consider x² + 4x - 437 = 0

It is difficult to make factors of x²+ 4x -437. In such a case factorization and hence the solution of quadratic equation can be found by the method of completing square and extracting square roots.

Examples of completing square method:

x² + 4x - 437 = 0

⇒ x² + 2(4/2)x = 347


Add (4/2) =(2)² to the both sides

x² + 4x + (2)² = 437 + (2)²

⇒ (x+2)² = 441

⇒ x+2 = ±√441 = ±21

⇒ x= ±21 - 2

x = 19 or x = -23

Hence solution set = {-23,19} 


By applying the quadratic formula:

Again there are some quadratic polynomials which are not factorable at all using integral coefficients. In such a case, we can always find the solution of a quadratic equation ax² + bx + c = 0 by applying a formula which is known as quadratic formula This formula is applicable for every quadratic equation.

it is:

 x= -b ±√b² -4ac /2a 


Derivation of quadratic formula:

Standard form of a quadratic equation is:

ax² + bx + c=0, a≠0


Step 1 Divide the equation by a

x² + b/a(x) + c/a = 0


Step 2 Take constant terms to the R.H.S

x² + b/a(x) = -c/a


Step 3 To complete the square on L.H.S. add (b/2a)² to the both sides


x² +b/a(x) +b²/4a² = b²/4a² - c/a

⇒[x+b/2a]² = b²-4ac/4a²

⇒ x+b/2a = ± √b² - 4ac/2a

⇒x= -b/2a ±√b² - 4ac/2a

x= -b±√b² - 4ac/2a


Hence the solution of the quadratic equation ax² +bx +c = 0 is given by

 x =-b±√b² - 4ac/2a 

which is called quadratic formula.

Example of applying the quadratic formula:


6x² + x - 15 = 0

a = 6 , b = 1 , c = -15

Putting the values in quadratic formula:

x=-b±√b² - 4ac/2a 

= -1 ±√(1)² - 4 (6)(-15)/ 2(6)

= -1 ±√361/ 12

= -1±19/12

i.e., x = -1+19/12     or x= -1-19/12

 x=3/2     or x= -5/3

Hence,😊

solution set = {3/2 , -5/2}

Post a Comment

0 Comments