Question

use the next three slides to factor $x^{2}-25$
$x^{2}-25=$
start by labeling the area model for rectangles i and iv. next,
multiply the main diagonal and enter the product into the table.

Explanation:

Step1: Recall the difference - of - squares formula

The formula for the difference of squares is \(a^{2}-b^{2}=(a + b)(a - b)\). In the expression \(x^{2}-25\), we can rewrite \(25\) as \(5^{2}\). So, \(a=x\) and \(b = 5\).

Step2: Apply the formula

Substitute \(a=x\) and \(b = 5\) into the formula \(a^{2}-b^{2}=(a + b)(a - b)\). We get \(x^{2}-25=(x + 5)(x - 5)\).

For the area - model:

• Rectangle I: If we consider \((x + 5)(x - 5)=x\times x+x\times(- 5)+5\times x+5\times(-5)\), rectangle I (with sides \(x\) and \(x\)) has an expression \(x^{2}\).
• Rectangle IV (with sides \(5\) and \(-5\)) has an expression \(-25\).
• The product of the main diagonal (rectangle I and rectangle IV) is \(x^{2}\times(-25)=-25x^{2}\).

Answer:

\(x^{2}-25=(x + 5)(x - 5)\); Rectangle I: \(x^{2}\), Rectangle IV: \(-25\), Product of Main Diagonal: \(-25x^{2}\)