Find the length and width of a rectangle with an area of x, squared, plus, 7, x, plus, 12x

Question

2
+7x+12 .
Answer
Attempt 1 out of 2
Multiple Choice Answers
left parenthesis, x, plus, 2, right parenthesis, left parenthesis, x, plus, 6, right parenthesis(x+2)(x+6)
left parenthesis, x, plus, 3, right parenthesis, left parenthesis, x, plus, 4, right parenthesis(x+3)(x+4)
left parenthesis, x, plus, 1, right parenthesis, left parenthesis, x, plus, 12, right parenthesis(x+1)(x+12)
left parenthesis, x, plus, 2, right parenthesis, left parenthesis, x, minus, 6, right parenthesis(x+2)(x−6)

Answer 1

To find the length and width of the rectangle based on the area given, we first need to factor the expression for the area, which is \( x^2 + 7x + 12 \).

We will factor the quadratic expression:

1. Look for two numbers that multiply to \( 12 \) (the constant term) and add up to \( 7 \) (the coefficient of \( x \)).
2. The numbers \( 3 \) and \( 4 \) satisfy this condition since \( 3 \cdot 4 = 12 \) and \( 3 + 4 = 7 \).

Now we can write the factored form of the expression:

\[ x^2 + 7x + 12 = (x + 3)(x + 4) \]

Thus, the length and width of the rectangle can be represented as:

Length: \( x + 4 \)

Width: \( x + 3 \)

So, the correct choice from the multiple choice answers is:

\[ \text{(x + 3)(x + 4)} \]