Question

A parallelogram has an area of x2+ 9x− 36.

a.What are expressions for the length and width of the parallelogram?

b.If x is an integer, what is the least possible value of x for a parallelogram to exist? Explain.

Answer 1

a) Length
`x + 12`, Width
`x - 3`, b)
`x = 3`

Explanation:

a) The expressions for the length and width of the parallelogram can be determined by factorizing the polynomial:

`A = x^(2) + 9\cdot x - 36`

`A = (x+12)\cdot (x-3)`

Length

`x + 12`

Width

`x - 3`

b) Since area is a positive variable, the set of solutions encompass all values of x such that area is zero o greater. Then, there are two interval where A is positive:

- When both binomials are negative.

`x \leq 3` and
`x \leq -12`

`x \leq -12`

- When both binomials are positive.

`x \geq 3` and
`x \geq -12`

`x \geq 3`

Since x is an integer, x must be positive and therefore the least possible value of x is:

`x = 3`