Question

1. reason a rectangle has an area of $x^{2}+7x + 12$ in.$^{2}$. use factoring to find possible dimensions of the rectangle. explain why you can use factoring to find the answer.

Explanation:

Step1: Factor the quadratic expression

For a quadratic expression \(ax^{2}+bx + c\) (here \(a = 1\), \(b=7\), \(c = 12\)), we need to find two numbers \(m\) and \(n\) such that \(m + n=b\) and \(m\times n=c\).
We look for two numbers that add up to \(7\) and multiply to \(12\). The numbers are \(3\) and \(4\) since \(3+4 = 7\) and \(3\times4=12\).
So, \(x^{2}+7x + 12=(x + 3)(x + 4)\)

Step2: Relate to the area formula of a rectangle

The area of a rectangle is \(A=\text{length}\times\text{width}\). If \(A=x^{2}+7x + 12\) and \(A=(x + 3)(x + 4)\) (from factoring), then the possible dimensions of the rectangle are \((x + 3)\) inches and \((x + 4)\) inches.
We can use factoring because the area formula \(A = lw\) (where \(l\) is length and \(w\) is width) for a rectangle is a product formula. When the area is given as a quadratic polynomial \(ax^{2}+bx + c\), factoring the polynomial gives us two binomials whose product is the area polynomial, and these binomials can be interpreted as the length and width of the rectangle.

Answer:

The possible dimensions of the rectangle are \((x + 3)\) inches and \((x + 4)\) inches.