A regular plot of land has an area of 240 square feet and is 8 feet longer than its width....

Question

a) Write a quadratic equation in the form ax² + bx + c = , whose solution gives the width of the rectangular plot of land.

The equation for the width is blank (Type an equation. Type your answer in standard form)

b) Solve the equation.

The solution(s) to the quadratic equation found in part a is/are x =

(Type an integer or a simplified fraction. Use a comma to separate answers as needed)

The width of the plot is blank (ft³, ft², or ft?)

Answer 1

The area of a rectangle is length•width. In this case, we know that it is 8 feet longer than its width, which means the length is 8 feet more than the width. So if the width is x, then we can write the length as x+8.

Now we can write length•width as x•(x+8). Multiplying, we get x² + 8x.

x² + 8x is the area, but we also know that the area is 240 square feet. So, x² + 8x = 240. Subtract 240 from both sides to get the equation:

x² + 8x - 240 = 0.

Now we need to solve the equation. We can do this by factoring, using the quadratic formula, or completing the square. I will show it by factoring:

To factor, we need to write x² + 8x - 240 in the form of (x + ___)(x + ____). The numbers in the blanks have to multiply to -240, and add to 8.

Here are some number pairs that multiply to -240:

-4 and 60

-24 and 10

-12 and 20

The numbers -12 and 20 are the ones we want, because they also add to 8. -12•20 = -240, and -12 + 20 = 8.

So we can factor x² + 8x - 240 as (x - 12)(x + 20). To solve, set each part equal to 0.

x - 12 = 0 means that x = 12

x + 20 = 0 means that x = -20

Both 12 and -20 are solutions to the equation, but they do not both make sense. The width of a rectangle cannot be negative, so the solution -20 does not work. The only solution that makes sense in context is that the width is 12.

To choose the units for width, think about what sort of shape "width" is. "Width" is a line. For lines, you use the basic units, not units squared or cubed. So the unit for width is just ft.

The width is 12ft.