Question

The area of a rectangular playing field is 192square meters. The length of the field is x+12 and the width is x-4. How do you calculate x by using quadratic formula?

Answer 1

`x = 12`

Explanation:

We know that the area formula for a rectangle is:

`"length" color(white)"." xx color(white)"." "width" color(white)"." = color(white)"." "area"`

So, we can plug these numbers in and then write everything in terms of a quadratic which we can solve with the quadratic formula.

`(x+12) xx (x-4) = 192`

Let's use the FOIL method to expand the left side.

`underbrace((x)(x)) _ "First" + underbrace((x)(-4)) _ "Outer" + underbrace((12)(x)) _ "Inner" + underbrace((12)(-4))_"Last" = 192`

`x^2 + (-4x) + (12x) + (-48) = 192`

`x^2 + 8x - 48 = 192`

Now subtract `192` from both sides.

`x^2 + 8x - 240 = 0`

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

This is a quadratic, so we can use the quadratic formula to solve it.

`a = 1`
`b = 8`
`c = -240`

`x = (-b+-sqrt(b^2-4ac))/(2a)`

Now plug in all of those values and simplify.

`x = (-(8)+-sqrt((8)^2-4(1)(-240)))/(2(1))`

`x = (-8+-sqrt(64+960))/2`

`x = (-8+-sqrt1024)/2`

Note that `1024 = 2^10 = (2^5)^2 = 32^2`

`x = (-8+-sqrt(32^2))/2`

`x = (-8+-32)/2`

`x = -4+-16`

This means our two values of `x` are:

`x = -4-16 " " and " " x = -4+16`

`x = -20 " " and " " x = 12`

Remember that `x` represents a length, and so it cannot possibly be negative. This leaves us with only one solution:

`x = 12`

Final Answer