Question

6.5 polynomial equations homework
score: 21.33/24 answered: 21/24
progress
question 22
0.33
part 1 of 2
the width of a rectangle is 4 feet more than its length.
if ( x ) represents the length of the rectangle, an expression giving the area is ( a=)
( x^{2}+4 x )
part 2 of 2
suppose the area of the rectangle is given by ( a=x(x + 4) ). if the area of the rectangle is 21 square feet,
then the length of the rectangle is feet and the width is
feet.
question help: video written example
submit question jump to answer

Explanation:

Step1: Set up the equation

Given \(A = x(x + 4)\) and \(A=21\). So, \(x(x + 4)=21\).
Expand the left - hand side: \(x^{2}+4x=21\).
Rearrange to get a quadratic equation: \(x^{2}+4x - 21=0\).

Step2: Solve the quadratic equation

For a quadratic equation \(ax^{2}+bx + c = 0\) (here \(a = 1\), \(b = 4\), \(c=-21\)), use the quadratic formula \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\).
First, calculate the discriminant \(\Delta=b^{2}-4ac=(4)^{2}-4\times1\times(-21)=16 + 84=100\).
Then \(x=\frac{-4\pm\sqrt{100}}{2}=\frac{-4\pm10}{2}\).
We have two solutions:
\(x_1=\frac{-4 + 10}{2}=\frac{6}{2}=3\) and \(x_2=\frac{-4-10}{2}=\frac{-14}{2}=-7\).
Since length \(x>0\), we take \(x = 3\).

Step3: Find the width

The width \(w=x + 4\). Substitute \(x = 3\) into \(w=x + 4\), we get \(w=3 + 4=7\).

Answer:

The length of the rectangle is \(3\) feet and the width is \(7\) feet.