From 11:00PM PDT on Friday, July 1 until 5:00AM PDT on Saturday, July 2, the Shmoop engineering elves will be making tweaks and improvements to the site. That means Shmoop will be unavailable for use during that time. Thanks for your patience!

# Number of Solutions to an Equation

### How Many Solutions Can an Equation Have?

Okay, algebristas,* different equations can have different numbers of solutions. Right now, we're only counting solutions from the real numbers. Some equations have solutions that are imaginary numbers, but we'll get to those later. No, you don't need to send us a Thank You card. Your words are enough.

*You know, people who serve mathuccinos. Alternatively, you.

### Sample Problems

• The equation x = 5 has only one solution: the number 5.
• The equation z2 = 4 has two solutions: z = 2 and z = -2.
• The equation x = x has infinitely many solutions: any value of x will work, since x is always equal to itself.
• The equation y2 = -5 has no real number solutions because the square of any real number is positive.

We interrupt this program to bring you a History Snack. Don't let it ruin your appetite.

Diophantus was a famous mathematician who's sometimes called "The Father of Algebra." Although with the way he got around, who wasn't he the father of? #ancientgossip

Anyway, D-man only liked positive rational solutions to equations. He would probably call a lot of the equations we'll be solving "absurd" since they have negative solutions. It's hard to blame him, though—after all, there's no such thing as a negative child. (He wishes. Oh, burn.)