A number is 20 less than its square. Find all answers.

The first step to solving this little algebra word problem is to create an equation out of it. Look for some words that indicate what type of actions you need to perform to solve this algebra problem.

The word is indicates the equals sign, and less than indicates subtraction. So the problem can be rewritten like this:

a number = its square – 20

If you choose the variable x to represent a number, then you end up with this equation:

x = x2 – 20

So you have a regular number, a variable, and that same variable squared. Hopefully, these numbers ring a bell. With just a little rearranging, you have a quadratic equation!

x2 – 20 = x

Now, just subtract x from both sides, and you're left with this:

x2 – x – 20 = 0

There are numerous ways to solve for a quadratic equation. The simplest way is probably solving by factoring. Begin the equation by creating two elements in parentheses and making x the first number in each element:

(x     )(x     ) = 0

Because the last operation in the quadratic equation is subtraction, you know that one of the elements must be addition, and the other must be subtraction, so that when you multiply the two last numbers, you get a negative number.

(x –  )(x +  ) = 0

Finally, you need to find two numbers whose product is –20 and whose sum is –1 (because –x is really –1x). The numbers 4 and 5 seem to fit the bill:

(x – 5)(x + 4) = 0

This is a good point to stop and quickly check your work. Use the FOIL method (First, Outer, Inner, Last) to multiply the two elements together and see if you get back to where you started. It looks like this:

  • First: x x x = x2
  • Outer: x x 4 = 4x
  • Inner: –5 x x = –5x
  • Last: –5 x 4 = –20

Add all those together and you get x2 + 4x – 5x – 20, or x2 –(1)x – 20 = 0, right back where you started!

Back to work! In order for (x – 5)(x + 4) to equal 0, one of the elements — either (x – 5) or (x + 4) — must equal zero. Set each of these equal to 0 and you'll have your answer:

  • If x – 5 = 0, then = 5
  • If x + 4 = 0, then x = –4

Now plug these answers into your original equation, x = x2 – 20, to check your answers:

  • (5) = (5)2 – 20
  • (–4) = (–4)2 – 20