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or, y(y 2 ) 3 (y 2 ) 0

or, (y 2 )(y 3 ) 0

? y 2 0 , or, y 3 0

If y 2 0 ,y 2

If y 3 0 ,y 3
Now, when y 2 ,

1

2





x a

x a

[putting the value of y]

or,
2 1

2 1





  

  

x a x a

x a x a

[by componendo and dividendo]

or,
1

3

2

2

a

x

or,x 3 a

Again, wheny 3 ,
1

3





x a

x a

or,
3 1

3 1





  

  

x a x a

x a x a

or,
2

4

2

2

a

x

or,
1

2

a

x

or,x 2 a
? Solution is :x 3 a, or, 2 a
Activity :

. Comparing the equation 1 x^2  1 0 with the equation ax^2 bxc 0 , write
down the values of a,b,c.
2. What is the degree of the equation (x 1 )^2 0? How many roots has the
equation and what are they?
5 ⋅6 Usage of quadratic equations
Many problems of our daily life can be solved easily by forming linear and quadratic
equations. Here, the formation of quadratic equations from the given conditions
based on real life problems and techniques for solving them are discussed.
Example 13. Denominator of a proper fraction is 4 more than the numerator. If the
fraction is squared, its denominator will be 40 more than the numerator. Find the
fraction.

Solution : Let the fraction be
x 4

x

.