Question

the rectangular area of the new school courtyard is represented by the expression: $4c^2 - 36$. which expression could represent the length times the width of the courtyard?

• $(2c - 6)^2$
• $(2c - 6)(2c + 6)$
• $4(c^2 - 32)$
• $(4c - 6)(c + 6)$

Explanation:

Step1: Recognize the difference of squares

The expression \(4c^2 - 36\) can be rewritten as \((2c)^2 - 6^2\). The difference of squares formula is \(a^2 - b^2=(a - b)(a + b)\).

Step2: Apply the difference of squares formula

Here, \(a = 2c\) and \(b = 6\). So, \((2c)^2 - 6^2=(2c - 6)(2c + 6)\).

Step3: Check other options (optional but for verification)

• For \((2c - 6)^2=(2c - 6)(2c - 6)=4c^2-24c + 36\), which is not equal to \(4c^2 - 36\).
• For \(4(c^2 - 32)=4c^2-128\), not equal to \(4c^2 - 36\).
• For \((4c - 6)(c + 6)=4c^2+24c-6c - 36=4c^2+18c - 36\), not equal to \(4c^2 - 36\).

Answer:

\((2c - 6)(2c + 6)\) (the second option)