Question

the width of a rectangle is the length minus 5 units. the area of the rectangle is 14 square units. what is the width, in units, of the rectangle?

Explanation:

Step1: Set up variables

Let the length of the rectangle be \( l \) units. Then the width \( w=l - 5 \) units. The area formula for a rectangle is \( A = l\times w \). Given \( A = 14 \), so \( l(l - 5)=14 \).

Step2: Expand the equation

Expand \( l(l - 5)=14 \) to get \( l^{2}-5l - 14=0 \).

Step3: Factor the quadratic equation

Factor \( l^{2}-5l - 14=0 \). We need two numbers that multiply to \(-14\) and add up to \(-5\). The numbers are \(-7\) and \(2\). So, \( (l - 7)(l+ 2)=0 \).

Step4: Solve for \( l \)

Set each factor equal to zero: \( l - 7=0\) gives \( l = 7\); \( l+2=0\) gives \( l=-2 \). Since length cannot be negative, \( l = 7 \).

Step5: Find the width

Since \( w=l - 5 \), substitute \( l = 7 \) into the formula. Then \( w=7 - 5=2 \).

Answer:

\(2\)