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?

1 Answer
Jul 14, 2017

Answer:

#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