Question

Given the expression for the area of the rectangle, factor to find possible expressions for the length and width. Area = 6x^2 + 11x - 2. What are the length and width?

Answer 1

To factor the expression for the area of the rectangle, \( 6x^2 + 11x - 2 \), we can start by looking for two numbers that multiply to \( 6 \times -2 = -12 \) (the product of the coefficient of \( x^2 \) and the constant term) and add up to \( 11 \) (the coefficient of \( x \)).

The two numbers that meet these criteria are \( 12 \) and \( -1 \). We can use these numbers to rewrite the middle term:

\[
6x^2 + 12x - 1x - 2
\]

Now we can group the terms:

\[
(6x^2 + 12x) + (-1x - 2)
\]

Factoring out the common terms in each group gives us:

\[
6x(x + 2) - 1(x + 2)
\]

Now we can factor out the common factor of \( (x + 2) \):

\[
(6x - 1)(x + 2)
\]

Thus, the factored form of the expression \( 6x^2 + 11x - 2 \) is:

\[
(6x - 1)(x + 2)
\]

In this case, we can assign the dimensions of the rectangle as:

- Length = \( 6x - 1 \)
- Width = \( x + 2 \)

So, the possible expressions for the length and the width are:

- Length = \( 6x - 1 \)
- Width = \( x + 2 \)