SOLVING ALGEBRAIC FRACTIONS INVOLVING QUADRATICS

To solve algebraic fractions, we can use the cross multiplication method.

Cross multiplication method :

5 × 14  =  10 × 7

70  =  70

Solve for x by first eliminating the algebraic fractions :

Example 1 :

4/x  =  x/2

Solution :

4/x  =  x/2

By cross multiplication, we get

 8  =  x²

x²  =  8

x  =  ± √8

So, the solution of x is ± √8

Example 2 :

x/5  =  2/x

Solution :

x/5  =  2/x

By cross multiplication, we get

x²  =  10

x  =  ± √10

So, the solution of x is ± √10

Example 3 :

(x-1)/4  =  3/x

Solution :

(x-1)/4  =  3/x

By cross multiplication, we get

x(x-1)  =  12

x² - x  =  12

x² – x - 12  =  0

By factorization, we get

(x – 4) (x + 3)  =  0

x  =  4 or -3

So, the solution of x is 4 or -3

Example 4 :

(x-1)/x  =  (x+11)/5

Solution :

(x-1)/x  =  (x+11)/5

By cross multiplication, we get

5(x-1)  =  x(x+11)

5x - 5  =  x² + 11x

x² + 11x - 5x + 5  =  0

x² + 6x + 5  =  0

By factorization, we get

(x + 1) (x + 5)  =  0

x  =  -1 or -5

So, the solution of x is -1 or -5

Example 5  :

x/(x+2)  =  1/x

Solution  :

x/(x+2)  =  1/x

By cross multiplication, we get

x²  =  x + 2

x² - x - 2  =  0

By factorization, we get

(x + 1) (x - 2)  =  0

x  =  -1 or 2

So, the solution of x is -1 or 2

Example 6  :

2x/(3x+1)  =  1/(x+2)

Solution  :

2x/(3x+1)  =  1/(x+2)

By cross multiplication, we get

2x(x + 2)  =  3x + 1

2x² + 4x  =  3x + 1

2x² + 4x – 3x – 1  =  0

2x² + x – 1  =  0

By factorization, we get

2x² + 2x – x – 1  =  0

2x(x + 1) – 1(x + 1)  =  0

(2x – 1) (x + 1)  =  0

x  =  1/2 or -1

So, the solution of x is 1/2 or -1

Example 7  :

(2x+1)/x  =  3x

Solution  :

(2x+1)/x  =  3x

By cross multiplication, we get

2x + 1  =  3x²

3x² – 2x – 1  =  0

By factorization, we get

3x² - 3x + x – 1  =  0

3x(x - 1) + 1(x - 1)  =  0

(3x + 1) (x - 1)  =  0

x  =  -1/3 or 1

So, the solution of x is -1/3 or 1

Example 8  :

(x+2)/(x–1)  =  x/2

Solution  :

(x+2)/(x–1)  =  x/2

By cross multiplication, we get

2(x + 2)  =  x(x - 1)

2x + 4  =  x² – x

x² – x – 2x – 4  =  0

x² – 3x – 4  =  0

By factorization, we get

(x – 4) (x + 1)  =  0

x  =  4 or -1

So, the solution of x is 4 or -1

After having gone through the stuff given above, we hope that the students would have understood, how to order fractions and decimals. 

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