CLASS-7ALGEBRAIC PROBLEM SOLUTION

Problem & Solution

Multiply every equations which are given below –

1)  (X + 4b) (X – 3a)  = ?

Ans.) (X + 4b) (X – 3a)

=  X. (X – 3a) + 4b. (X – 3a)

=  X² - 3aX + 4bX – 12ab       (Ans.)

2) ( X²- 4 ) ( X²- 5 ) + X ( X + 8 ) =  ?

Ans.) ( X² - 4 ) ( X² - 5 ) + X ( X + 8 )

=  X². ( X² - 5 ) –  4 . ( X² - 5 ) +  X . X + 8X

=  X². X² - 5 X² - 4 X² + 20 + X¹⁺¹ + 8X

=   X²⁺² - 9X² + 20 + X² + 8X

=  X⁴ - ( 9 – 1) X² + 8X + 20

=  X⁴ - 8X² + 8X + 20          (Ans.)

3) ( X + 5 ) ( X – 6 ) ( X – 4 ) + Xᶟ - 4 X² + 3 X – 20  =  ?

Ans.)  ( X + 5 ) ( X – 6 ) ( X – 4 ) + X - 4 X² + 3 X – 20

= { X . ( X – 6 ) + 5 . ( X – 6 ) } ( X – 4 ) + X - 4 X² + 3 X – 20

= {( X² - 6X + 5X – 30 ) ( X – 4 )} + X - 4 X² + 3 X – 20

= { X² - ( 6 – 5 ) X – 30 } ( X – 4 ) + X - 4 X² + 3 X – 20

= ( X² - X – 30 ) ( X – 4 ) + X - 4 X² + 3 X – 20

= X² ( X – 4 ) – X ( X – 4 ) – 30 ( X – 4 ) +  X - 4 X² + 3 X – 20

= X²⁺¹ - 4X² - X¹⁺¹ + 4X – 30X + 120 + X - 4 X² + 3 X – 20

=  X- 4X² - X² - 26X + 120 + X - 4X² + 3 X – 20

=  ( 1 + 1 ) X- ( 4 + 1 + 4 ) X² - ( 26 – 3 ) X + 120 – 20

=  2 X- 9 X² - 23 X + 120 – 20

=  2X- 9X² - 23X + 100                (Ans.)

Simplify the following using a special product

1)  ( 2X – 5a )²  = ?

Ans.)     ( 2X – 5a )²

=   ( 2X – 5a ) ( 2X – 5a )

=  2X ( 2X – 5a ) - 5a ( 2X – 5a )

=  2.2 X¹⁺¹- 2X.5a - 5a.2X + 5a.5a

=  4X²- 10aX - 10aX + 25a²

=  4X²- 20aX + 25a²

We can get the same result via applying formula –

( a – b )² = a² - 2ab + b²

So, as per the given condition with applying formula

( 2X – 5a )²

=  (2X)²- 2 (2X. 5a) + 25a²

=  4X²- 20aX + 25a²           (Ans.)

2)  ( 3x + 5b )²  = ?

Ans.)     ( 3x + 5b )²

=  ( 3x + 5b ) ( 3x + 5b )

=  3x. ( 3x + 5b ) + 5b ( 3x + 5b )

=  3.3 x¹⁺¹ + 3x.5b + 5b.3x + 5.5 b¹⁺¹

=  9x² + 15xb + 15xb + 25b²

=  9x² + 30xb + 25b²          (Ans.

We can get the same result via applying formula –

( a + b )² = a² + 2ab + b²

So, as per the given condition with applying formula

( 3x + 5b )²

=  (3x)² + 2 (3x.5b) + (5b)²

=  9x² + 30xb + 25b²              (Ans.)

3)   ( 3X – 1 / 2X)²  = ?

1

(   3X  -  ---------  ) ²

2X

1                            1

=   ( 3X -  ---------  )  .  ( 3X  -  --------  )

2X                          2X

1              1                   1

= 3X . ( 3X  - ------- ) - -------- . ( 3X - ------- )

2X            2X                  2X

3X           3X               1

= 3X . 3X -  -------- - --------- +  -----------

2X            2X           2X . 2X

3             3            1

=  9X² -  -------  -  ------- +  -------

2             2           4X²

1

=   9X² + --------  -  3               (Ans.)

4X²

We can get the same result via applying formula –

( a – b )² = a² - 2ab + b²

So, as per the given condition with applying formula

1

( 3X - ------ ) ²

2X

1              1²

= ( 3X )² - 2 . 3X . ------- +  ( -------- )

2X             2X²

1

=  9X² - 3 + --------

4X²

1

= 9X² + -------  - 3          (Ans.)

4X²

Find the square of each of the following binomial –

2²               0               25

=  ---------  -  ---------  +  ---------

X²              XY               Y²

Express each of the following equations as the square of a binomial

1) a² - 16ab + 64b²

Ans.)    a² - 16ab + 64b²

=  a² - 2a . 8b +  (8b)²

=   ( a – 8b )²           (Ans.)

2)   25a² + 40ab + 16b²

Ans.)   25a²  + 40ab + 16b²

=   (5a)² + 2. 5a . 4b +  (4b)²

=   ( 5a + 4b )²         (Ans.)

Find the value of the following using special products –

1)  199² = ?

Ans.)    199²  =  ( 200 – 1)²

=   200² - 2 . 200 . 1 + 1²

=   40000 – 400 + 1

=   39600 + 1

=   39601

2)   310² = ?

310² =  ( 300 + 10 )²

=   300² + 2. 300 . 10 + 10²

=  90000 + 6000 + 100

=  96100                (Ans.)

3)   4.5 X 5.5 = ?

4.5 X 5.5   =  ( 5.00 – 0.50 ) X ( 5.00 + 0.50 )

=  ( 5.00 )² - ( 0.50 )²

=   25 -0.25

=  24.75           (Ans.)

4)   3.6 X  4.4  = ?

3.6 X  4.4  ( 4.00 – 0.40 ) X ( 4.00 + 0.40 )

=  ( 4.00 )² - ( 0.40 )²

=  16 - 0.16

=  15.84            (Ans.)

5)   10.2 X 9.8 =  ?

10.2 X 9.8  ( 10.00 + 0.20 ) X ( 10.00 – 0.20 )

=   ( 10.00 )² - ( 0.20 )²

=   100 – 0.04

=   99.96        (Ans.)