Question

known for their artistic weaving ability, the diné (commonly known as navajo) create durable blankets with vivid patterns. the quadratic expression ((x + 2)^2 - 4) is one way to represent the area of a rectangular diné blanket, where (x) represents the width of the blanket, in feet. a partially complete equivalent form of the quadratic expression is shown below. determine the equivalent form of the expression. then, use the drop-down menus to explain the meaning of the equivalent form. equivalent form: (x (square)) click the arrows to choose an answer from each menu. the completed equivalent form of the quadratic expression is choose... this equivalent form represents choose... of the blanket. the width of the blanket is choose... and the length is choose... the width.

Explanation:

Step1: Expand the given expression

We start with the expression \((x + 2)^2-4\). First, expand \((x + 2)^2\) using the formula \((a + b)^2=a^2+2ab + b^2\), where \(a=x\) and \(b = 2\). So \((x + 2)^2=x^{2}+4x + 4\). Then subtract 4: \(x^{2}+4x+4 - 4=x^{2}+4x\).

Step2: Factor the expanded expression

Now, factor \(x^{2}+4x\). We can factor out an \(x\) from each term. So \(x^{2}+4x=x(x + 4)\).

Answer:

The equivalent form of the quadratic expression \((x + 2)^2-4\) is \(x(x + 4)\). So the value in the box is \(x + 4\).

For the meaning part:

• The completed equivalent form of the quadratic expression is \(x(x + 4)\).
• This equivalent form represents the area (since the original expression represents the area of the blanket) of the blanket.
• The width of the blanket is \(x\) (given that \(x\) represents the width) and the length is \(x + 4\) (since area of a rectangle is width \(\times\) length, and we have factored the area as \(x\times(x + 4)\)) the width.