Question

candy draws a square design with a side length of x inches for the window at the pet shop. she takes the design to the printer and asks for a sign that has an area of $16x^2 - 40x + 25$ square inches.
what is the side length, in inches, of the pet shop sign?
$\bigcirc$ $4x + 5$
$\bigcirc$ $4x - 5$
$\bigcirc$ $8x + 5$
$\bigcirc$ $8x - 5$

Explanation:

Step1: Recall the perfect square trinomial formula

The formula for a perfect square trinomial is \((a - b)^2 = a^2 - 2ab + b^2\). We need to factor the given area expression \(16x^2 - 40x + 25\) to find the side length of the square (since the area of a square is side length squared).

Step2: Identify \(a\) and \(b\) in the perfect square formula

First, look at the first term \(16x^2\). We can write this as \((4x)^2\), so \(a = 4x\).
Next, look at the last term \(25\). We can write this as \(5^2\), so \(b = 5\).
Now, check the middle term. The middle term of the perfect square trinomial should be \(-2ab\). Let's calculate \(-2ab\) with \(a = 4x\) and \(b = 5\):
\(-2ab=-2\times(4x)\times5=-40x\), which matches the middle term of the given expression \(16x^2 - 40x + 25\).

Step3: Factor the expression

Using the perfect square trinomial formula, \(16x^2 - 40x + 25=(4x - 5)^2\). Since the area of the square is \((\text{side length})^2\), the side length is \(4x - 5\).

Answer:

\(4x - 5\) (corresponding to the option: \(4x - 5\))