Question

Study the products shown. Is there a pattern?

(x + 3)2 = x2 + 6x + 9

(x + 4)2 = x2 + 8x + 16

(x + 5)2 = x2 + 10x + 25

(x + 6)2 = x2 + 12x + 36

Each product is in the form ax2 + bx + c.

Which of the following describes the relationship between b and c?

c is 1.5 times that of b.

c is double b.

c is the square of half of b.

c is the square of b.

Answer 1

c is the square of half of b.

This is evident in all products,

(6/2)² = 3²
= 9
(8/2)² = 4²
= 16
(10/2)² = 5²
= 25
(12/2)² = 6²
= 36
(b/2)² = c

Answer 2

The correct option is 3. c is the square of half of b.

Explanation:

The given equations are

`(x+3)^2=x^2+6x+9`

`(x+4)^2=x^2+8x+16`

`(x+5)^2=x^2+10x+25`

`(x+6)^2=x^2+12x+36`

These are the perfect square. According to this pattern,

`(x+y)^2=x^2+2xy+y^2` .... (1)

Each product is in the form

`(x+y)^2=ax^2+bx+c` .... (2)

From (1) and (2), we get

`b=2y`

`c=y^2`

`c=((b)/(2))^2`
`[\because b=2y]`

It means c is the square of half of b.

Therefore the correct option is 3.