Question

6. What term must be added to each of the following
to make it a perfect square?
a. 25 - 30p
b. 64t^2 + 121u^2
c. 49m^2- 84m
d. 81s^2_90st
e. 1664 + m^4
​

Answer 1

Given,

The terms are

1. 25-30p
2. 64t²+121u²
3. 49m²-84m
4. 81s²-90st
5. 1664+m⁴

To find,

The terms which should be added to the given terms, to make them perfect squares.

Solution,

We can simply solve this mathematical problem by using the following mathematical process.

Here, we have to convert the given terms into the format of,

Either (a²+2ab+b²) or (a²-2ab+b²) by adding a term. This will eventually produce (a+b)² or (a-b)², which are perfect squares. These terms can be guessed easily by observing the given term, and finding out which term is missing from a²,2ab and b².

1) 25 -30p = (5)²-(2×5×3p)

Here, we have to add = (3p)² = 9p²

Then,

(5)²-(2×5×3p)+9p²

= (5)²-(2×5×3p)+(3p)²

= (5-3p)²

2) 64t²+121u² = (8t)²+(11u²)

Here, we have to add = (2×8t×11u) = 176tu

Then,

(8t)²+176tu+(11u²)

= (8t)²+(2×8t×11u)+(11u²)

= (8t+11u)²

3) 49m²-84m = (7m)²-(2×7m×6)

Here, we have to add = (6)² = 36

Then,

(7m)²-(2×7m×6)+(6)²

= (7m-6)²

4) 81s²-90st = (9s)²-(2×9s×5t)

Here, we have to add = (5t)² = 25t²

Then,

(9s)²-(2×9s×5t)+(5t)²

= (9s-5t)²

5) 1664+m⁴ = 1664+(m²)²

Here, we have to add = 100+(2×42×m²) = 100+84m² (Here, we added 100 because 1664 is not a perfect square, the nearest perfect square is [more in value] 1764)

Then,

1664+(m²)²+100+84m²

= 1764+84m²+(m²)²

= (42)²+(2×42×m²)+(m²)²

= (42+m²)²

Hence, we have to add 9p²,176tu,36,25t²,(100+84m²) respectively.

Answer 2

Answer:Given,

The terms are

25-30p

64t²+121u²

49m²-84m

81s²-90st

1664+m⁴

To find,

The terms which should be added to the given terms, to make them perfect squares.

Solution,

We can simply solve this mathematical problem by using the following mathematical process.

Here, we have to convert the given terms into the format of,

Either (a²+2ab+b²) or (a²-2ab+b²) by adding a term. This will eventually produce (a+b)² or (a-b)², which are perfect squares. These terms can be guessed easily by observing the given term, and finding out which term is missing from a²,2ab and b².

1) 25 -30p = (5)²-(2×5×3p)

Here, we have to add = (3p)² = 9p²

Then,

(5)²-(2×5×3p)+9p²

= (5)²-(2×5×3p)+(3p)²

= (5-3p)²

2) 64t²+121u² = (8t)²+(11u²)

Here, we have to add = (2×8t×11u) = 176tu

Then,

(8t)²+176tu+(11u²)

= (8t)²+(2×8t×11u)+(11u²)

= (8t+11u)²

3) 49m²-84m = (7m)²-(2×7m×6)

Here, we have to add = (6)² = 36

Then,

(7m)²-(2×7m×6)+(6)²

= (7m-6)²

4) 81s²-90st = (9s)²-(2×9s×5t)

Here, we have to add = (5t)² = 25t²

Then,

(9s)²-(2×9s×5t)+(5t)²

= (9s-5t)²

5) 1664+m⁴ = 1664+(m²)²

Here, we have to add = 100+(2×42×m²) = 100+84m² (Here, we added 100 because 1664 is not a perfect square, the nearest perfect square is [more in value] 1764)

Then,

1664+(m²)²+100+84m²

= 1764+84m²+(m²)²

= (42)²+(2×42×m²)+(m²)²

= (42+m²)²

Step-by-step explanation: