Question

question 3
the area of a rectangle is represented by the expression ( 64x^{2}-1 )
which of the following could represent the dimensions of the rectangle?
( (32x - 1)(2x + 1) )
( (64x + 1)(64x - 1) )
( (16x - 1)(4x + 1) )
( (8x + 1)(8x - 1) )

Explanation:

Step1: Recall the difference - of - squares formula

The formula for the difference of squares is \(a^{2}-b^{2}=(a + b)(a - b)\).
For the expression \(64x^{2}-1\), we can rewrite it as \((8x)^{2}-1^{2}\), where \(a = 8x\) and \(b=1\).

Step2: Apply the difference - of - squares formula

Using the formula \(a^{2}-b^{2}=(a + b)(a - b)\), when \(a = 8x\) and \(b = 1\), we get \((8x)^{2}-1^{2}=(8x + 1)(8x-1)\).

Let's check the other options:

• For \((32x - 1)(2x + 1)=32x\times2x+32x\times1-1\times2x - 1\times1=64x^{2}+32x-2x - 1=64x^{2}+30x - 1

eq64x^{2}-1\).

• For \((64x + 1)(64x - 1)=(64x)^{2}-1^{2}=4096x^{2}-1

eq64x^{2}-1\).

• For \((16x - 1)(4x + 1)=16x\times4x+16x\times1-1\times4x - 1\times1=64x^{2}+16x-4x - 1=64x^{2}+12x - 1

eq64x^{2}-1\).

Answer:

\((8x + 1)(8x - 1)\)