FINDING THE ROOTS OF A QUADRATIC EQUATION EXAMPLES

Example 1 :

Find the roots of the following equations:

x - (1/x)  =  3,  x ≠ 0

Solution :

(x² - 1)/x  =  3

x² - 1  =  3 x

x² - 3 x - 1  =  0

a  =  1, b  = -3, c = -1

x  =  - b ± √(b² - 4 a c)/2a

x  =  - (-3) ± √((-3)² - 4 (1)(-1))/2(1)

x  =  3 ± √(9+4)/2

x  =  (3 ± √13)/2

Example 2 :

Find the roots of the following equations:

[1/(x + 4)] - [1/(x -7)]  =  11/30

Solution :

[1/(x + 4)] - [1/(x -7)]  =  11/30

[(x - 7) - (x + 4)]/(x + 4)(x - 7)  =  11/30

(x - 7 - x - 4)/(x² + 4 x - 7 x - 28)  =  11/30

-11/(x² - 3 x - 28)  =  11/30

30 (-11)  =  11 (x² - 3 x - 28)

-30  =  x² - 3 x - 28

x² - 3 x - 28 + 30  =  0

x² - 3 x + 2  =  0

x² - 1 x - 2 x + 2  =  0

x (x - 1) - 2 (x - 1)  =  0

(x - 2) (x - 1)  =  0

 x - 2 = 0        x - 1 = 0

  x = 2       x = 1

Example 3 :

The sum of the reciprocals of Rehman's ages (in years) 3 years ago and 5 years from now is 1/3. Find his present age.

Solution :

Let 'x' be Rehman's present age

Three years ago his age was x - 3

5 years after his age will be x + 5

Sum of the reciprocals of Rehman's ages  =  1/3

[1/(x -3) + 1/(x + 5)]  =  1/3

(x + 5 + x - 3)/(x-3)(x + 5)  =  1/3

(2 x + 2)/(x² - 3 x + 5 x - 15)  =  1/3

(2 x + 2)/(x² + 2 x - 15)  =  1/3

3 (2 x + 2)  =  1 (x² + 2 x - 15)

6 x + 6  =  x² + 2 x - 15

x² + 2 x - 15 - 6 x - 6  =  0

x²- 4 x - 21  =  0

x² - 7 x + 3 x - 21  =  0

x (x - 7) + 3 (x - 7)  =  0

(x + 3) (x - 7)  =  0

 x  =  3  or x  =  7

So his present age will be 7 years.

Example 4 :

In a class test,the sum of Shefali's marks in Mathematics and English is 30. Had she got 2 marks more in Mathematics and 3 marks less in English, the product of their marks would have been 210. Find her marks in the two subjects.

Solution : 

Let 'x' be the mark got by Shefali in the subject mathematics

let '30 - x' be the mark got by Shefali in English

(x + 2)(30 - 3 - x)  =  210

(x + 2)(27 - x)  =  210

27 x - x² + 54 - 2 x - 210  =  0

- x² + 25 x - 156  =  0

x² - 25 x + 156  =  0

x² - 12 x - 13 x + 156  =  0

x (x - 12) - 13 (x - 12)  =  0

(x - 12) (x - 13)  =  0

x - 12  =  0          x - 13  =  0

x  =  12           x  =  13

If Shefali's mathematics marks  =  12 then her English mark will be = 30 - 12 = 18. 

If Shefali's mathematics marks = 13 then her English mark will be = 30 - 13 = 17.

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