Question

A rectangular floor area can be completely tiled with 200 square tiles. If the side length of each tile is increased by 1 unit, it would take only 128 tiles to cover the floor.

1. Assuming the original length of each side of a tile be x units, make a quadratic equation from the above information.   [1]
2. Write the corresponding quadratic equation in standard form.   [1]
3.    
   1. Find the value of x, the length of side of a tile by factorisation.   [2]
                                       OR
   2. Solve the quadratic equation for x, using quadratic formula.   [2]

Answer 1

i. Let the original side length of each tile be x units.

The area of the rectangular floor using 200 tiles = 200 x² unit² 

The area with increased side length (each side increased by 1 unit) using 128 tiles

= 128 (x + 1)² unit²

So, the required quadratic equation is:

200x² = 128 (x + 1 )²

ii. We have,

200x² = 128 (x + 1)²

⇒ 200x² = 128 (x² + 2x + 1)

⇒ 200x² = 128x² + 256x + 128

⇒ 72x² − 256x − 128 = 0

which is the quadratic equation is standard form.

iii. 
a. We have,

72x² − 256x − 128 = 0

or, 9x² − 32x − 16 = 0

or, 9x² − 36x + 4x − 16 = 0

or, 9x (x − 4) + 4 (x − 4) = 0

or, (x − 4) (9x + 4) = 0

or, x = 4, `-4/9`

Since the side cannot be negative, thus x = 4 units.

                                                   OR

b. We have 9x² − 32x − 16 = 0

On comparing with ax² + bx + c = 0, we get

a = 9, b = −32 and c = −16

Using quadratic formula,

`x = (-b +-sqrt(b^2 - 4ac))/(2a)`

`therefore x = (-(-32) +-sqrt((-32)^2 - 4(9)(-16)))/(2 xx 9)`

= `(32 +-sqrt(1024 + 576))/(18)`

= `(32 +-sqrt(1600))/(18)`

= `(32 +- 40)/18`

= `(32 + 40)/18 or (32 - 40)/18`

= `72/18 or (-8)/18 = 4 or (-4)/9`