Question

quadratic word problems
the area of a rectangle is 20 cm². the length is 8 more than the width. use w for width.
what is the width in cm?
equation

Explanation:

Step1: Define variables and length

Let the width be \( w \) cm. Then the length is \( w + 8 \) cm (since length is 8 more than width).

Step2: Use area formula for rectangle

The area of a rectangle is \( \text{length} \times \text{width} \). So we set up the equation: \( w(w + 8)=20 \)
Expand the left side: \( w^{2}+8w = 20 \)
Rearrange to standard quadratic form: \( w^{2}+8w - 20=0 \)

Step3: Solve the quadratic equation

We can solve this quadratic equation using the quadratic formula \( w=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a} \) for \( ax^{2}+bx + c = 0 \). Here, \( a = 1 \), \( b = 8 \), \( c=- 20 \).
First, calculate the discriminant \( \Delta=b^{2}-4ac=(8)^{2}-4\times1\times(-20)=64 + 80 = 144 \)
Then, \( w=\frac{-8\pm\sqrt{144}}{2\times1}=\frac{-8\pm12}{2} \)
We have two solutions:
\( w_{1}=\frac{-8 + 12}{2}=\frac{4}{2}=2 \)
\( w_{2}=\frac{-8-12}{2}=\frac{-20}{2}=- 10 \)
Since width cannot be negative, we discard \( w=-10 \).

Answer:

The width is \( 2 \) cm.