A number is 12 less than its square. Find all such numbers.

Question

Solve the problem. use the five-step problem-solving approach.

 

Answer 1

There are multiple "five-step approaches."  If you are learning a different one, then change this to fit it.

 

1. Think; underline important information

     A number is 12 less than its square. Find all such numbers. 

     You may read and re-read the problem until you understand it and can put it into your own words.  How's this:  Find all the numbers that are 12 less than the number squared.

    You will have to multiply a number times itself (square) and subtract.  You need one variable, so let's call the number x.

 

2. Figure it out.

    Let's translate the word (story) problem into a math equation (with expressions having variables, constants, and operations):

    "A number is 12 less than its square"      means

          x         =   -12        +      x * x   

                (note:  "is" means equals; "less than" means subtract; "square" means times itself)

 

3. Come up with a number scheme

    Let's re-arrange  the equation in a way that helps us find all the possible values for x.  Since there is both an x and x*x, let's put them together on one side of the equal sign and see what we get.

 

4.  Solve the problem

          x = -12 + x*x         (re-write)

         x - x*x = -12           (subtract x*x from each side)

         -x*x + x = -12         (re-arrange terms)

        x*x - x = 12             (multiply by -1 to change the minus signs)

         x*(x-1) = 12            (factor)

              [aha! a number times one less than the number = 12; I can understand that much better !]

      The answer must be x=4, because 4*(4-1) = 4*3 =12

          And, if negative numbers are allowed, x = -3, because (-3)(-4) = 12.

       That;s all such numbers.

 

5.   Check and Discuss

     The math wasn't too hard.      4*3=12  and (-3)*(-4)=12

     The problem was much easier to understand and solve after I wrote it with factors (two terms multiplied).

     If negative numbers are allowed, it is pretty easy to forget about them and just give one answer for x.  Gotta watch that !

Answer 2

n represents the number

 

n = -12 + n²        reorder terms

n = n² - 12          subtract n from both sides

0 = n² - n - 12     FOIL

0 = (n - 4)(n + 3)

 

{-3, 4}

Answer 3

Hello Nikki!

 

I'm not sure what "the five-step problem-solving approach" is, but I can tell you that the best way to solve this - or any - word problem is to deconstruct it into a series of equations that can be manipulated more easily. Let's analyze the question:

 

"A number..." - so we're finding an unknown. Let's call it N, for "number."

"...is 12 less than..." - we are subtracting 12 from something to get N (i.e. 'something' - 12 = N)

"...than its [N's] square." - that 'something' I mentioned in the last line is the square of N, or N^2.

 

Put it all together, and what you get is an equation:

 

N^2 - 12 = N

 

So now, the goal is to solve for N. The first thing I would do would be to subtract N from the equation (remember to apply this to both sides of the equals sign) and you will get:

 

N^2 - N - 12 = 0

 

This is what is called a "quadratic equation," and can be solved generally using an equation most like to call "the quadratic formula." However, in many cases, the equation can be factored. In this case,

 

N^2 - N - 12 = (N - 4)*(N - 3)

 

Factoring that was second nature to me because I've been doing this for a while. Examining whether an equation can be factored takes time and experience. How did I know to use 4 and 3? Let's take a look at the equation... I'm assuming you are familiar with the "FOIL" acronym (First, Outside, Inside, Last). FOIL is an excellent mnemonic device to remember how multiplication is carried out.

 

(N - 4) * (N + 3) = N*N (first) + 3*N (outside) - 4*N (inside) - 4*3 (last)

 

I chose -4 and +3 because -4*+3 gives you -12, and if you add 3*N to -4*N, you get -N, which is exactly what we needed. Like I said: Time and experience. Look at the only other alternatives: to get -12 you could either use -12 and +1 (which would get you -11*N for the middle term), -6 and +2 (which would yield -4*N in the middle term), or -4 and +3, which it turns out was what we needed.

 

Anyway, back to the main problem. We are left with

 

(N-4)*(N+3) = 0

 

which only has two possible solutions: N=4, or N=-3

 

Let's check and see if this works.

 

"-3 is 12 less than its square" implies that "9-12=-3", which it does. CHECK!

"+4 is 12 less than its square" implies that "16-12 = 4", which it does. YAY!

 

Okay, so problem solved. :)

 

I hope that helped!