Since we know that the area of the rectangle is 60 sq. ft, we know how it would be represented given our length and width:
This is because, like any rectangle, length x width=area, even if we're not working with actual numbers and polynomials instead.
Now, we need to solve for x. To do this, we need to multiply both of the binomials together. Using FOIL, we'd get the following:
x2 - 4x + 3x - 12 =60
Combine the two middle terms, and we have:
x2 - x - 12 =60
Now, this could be considered the quadratic equation that would represent the situation.
Next, we could make this into a quadratic equation that we can work with to solve for x. TO do that, I'll subtract 60 from both sides to give us this:
x2 - x - 72 = 0
Now, we need to use factoring to figure out what x could possibly be. Factoring is a strategy used for solving quadratic equations in which we work to find two numbers, which I'll call a and b. a and b must have the following properties:
a+b= middle term of our quadratic equation
a*b=our contstant term (the last term in the equation)
In our case, this means that:
To find these two numbers, we'll take factors of 72 and see which combination would give us a sum of -1.
72 1 -> nope
36 2 ->no
24 3 -> still too big
12 6-> still too big
9 8-> yes! If we have +8 and -9, their sum is -1!
Ok, now that we have our numbers, we can now write our equation in factored form:
So, we have two solutions. Which one is right? Well, keep in mind the following when it comes to factoring quadratic equations: when you factor it, the value of x that will actually make the equation 0 is the opposite of the sign that you see in the factor. What do I mean by that? Basically the following:
since (x-9) is a factor, x=9 is a zero/root of the quadratic equation. A value of x=9 would make the quadratic equation 0.
since (x+8) is a factor, x=-8 is a zero/root of the quadratic equation. A value of x=-8 would make the quadratic equation 0.
Plug it in yourself and you'll see that it works like this.
So while we have (x-9) and (x+8) as factors of the equation, the real x values we need to think about are x=9 and x=-8. Let's go back to our original rectangle. We know that the length was (x+3) and the width was (x-4). Let's see what happens if we plug in our new x values
Length= (-8+3) = -5
Width= (-8-4)= -12
-5*-12= 60. While this does work, we can't such a thing as -5 ft for the length of a garden! Thus, we can't count this solution-it's what we call an extraneous solution. Let's try 9 now:
Length= (9+3) = 12
Width = (9-4) = 5
12*5= 60. This works because both the length and width are still positive, and we multiply it out to 60.
So there you have it- x=9! I hope this helps, and be sure to review how to factor a quadratic equation as shown.
Hope this helps!