Question

josephine has a rectangular garden with an area of $2x^2 + x - 6$ square feet. which expressions can represent the length and width of the garden?\
\bigcirc length = $x^2 - 3$ feet; width = 2 feet\
\bigcirc length = $2x + 3$ feet; width = $x - 2$ feet\
\bigcirc length = $2x + 2$ feet; width = $x - 3$ feet\
\bigcirc length = $2x - 3$ feet; width = $x + 2$ feet

Explanation:

Step1: Recall the area formula for a rectangle

The area \( A \) of a rectangle is given by \( A = \text{length} \times \text{width} \). Here, the area is \( 2x^2 + x - 6 \), so we need to factor this quadratic expression to find the length and width.

Step2: Factor the quadratic expression \( 2x^2 + x - 6 \)

We use the method of factoring by grouping. For a quadratic \( ax^2 + bx + c \) (here \( a = 2 \), \( b = 1 \), \( c = -6 \)), we need two numbers that multiply to \( a \times c = 2 \times (-6)= -12 \) and add up to \( b = 1 \). The numbers are \( 4 \) and \( -3 \) (since \( 4 \times (-3)= -12 \) and \( 4 + (-3)=1 \)).

Rewrite the middle term using these numbers:
\( 2x^2 + 4x - 3x - 6 \)

Group the terms:
\( (2x^2 + 4x) + (-3x - 6) \)

Factor out the greatest common factor from each group:
\( 2x(x + 2) - 3(x + 2) \)

Now, factor out the common binomial factor \( (x + 2) \):
\( (2x - 3)(x + 2) \) Wait, no, wait. Wait, let's check again. Wait, if we have \( 2x(x + 2)-3(x + 2) \), that is \( (2x - 3)(x + 2) \)? Wait, no, that would give \( 2x^2 +4x -3x -6=2x^2 +x -6 \), which is correct. Wait, but let's check the options. Wait, the options have \( 2x - 3 \) and \( x + 2 \)? Wait, no, the options: let's look again. The options are:

1. length \( = x^2 - 3 \), width \( = 2 \)
2. length \( = 2x + 3 \), width \( = x - 2 \)
3. length \( = 2x + 2 \), width \( = x - 3 \)
4. length \( = 2x - 3 \), width \( = x + 2 \)

Wait, when we factor \( 2x^2 +x -6 \), let's do it again. Let's use the AC method correctly. \( a = 2 \), \( c = -6 \), \( ac = -12 \). We need two numbers that multiply to -12 and add to 1. The numbers are 4 and -3 (4 * (-3) = -12, 4 + (-3)=1). So:

\( 2x^2 + 4x - 3x - 6 = 2x(x + 2) - 3(x + 2) = (2x - 3)(x + 2) \). Yes, that's correct. So the factors are \( (2x - 3) \) and \( (x + 2) \). So length and width can be \( 2x - 3 \) feet and \( x + 2 \) feet (or vice versa, since multiplication is commutative). Looking at the options, the fourth option is "length \( = 2x - 3 \) feet; width \( = x + 2 \) feet".

Answer:

length \( = 2x - 3 \) feet; width \( = x + 2 \) feet (the fourth option)