We call equations like $$x−7=2 linear equations. These are equations where all variables have a power of $$1. Linear equations have only one solution. That is, there is only one value of the variable which will make the equation true. In this case, the only solution is $$x=9
If instead, the variables in the equation have a power of $$2, we call them quadratic equations. Quadratic equations can potentially have two solutions. For example, in the equation $$x2−7=2, there are two solutions, $$x=3 and $$x=−3.
Exploration
Solve $$x2+4=20 for $$x.
Following our rules for solving linear equations, we want to isolate $$x and whatever we do to one side of the equation we do to the other. So our first step is to subtract $$4 from both sides of the equation, giving us $$x2=16.
Next we want to undo raising $$x to the power of $$2. However, we need to be careful here. There are two operations which could be the reverse of squaring a number. We have to take both the positive and negative square root. This will give us two solutions.
| $$x2 | $$= | $$16 | |
| $$x | $$= | $$±√16 |
Taking the positive and negative square root of both sides |
| $$= | $$±4 |
Evaluating the positive and negative square roots |
The symbol $$± means "plus or minus". We can use this as a shorthand for both the positive and negative of a number. In this case, it means that our solutions are $$x=4 and $$x=−4.
We can check these solutions by substituting them in to the original equation and seeing if it holds true.
We call raising a number to the power of $$2 "squaring" the number. "Quadratic" comes from the ancient Latin for "square". So quadratic equations can be thought of as square equations.
Equations where all of the variables have a power of one are linear equations. These have at most one solution.
Equations where some of the variables have a power of two are quadratic equations. These have at most two solutions.
The symbol $$± means "plus or minus".
If we can rearrange a quadratic equation into the form $$x2=k, then we can solve the equation by taking the positive and negative square roots. That is, $$x=±√k.
Practice questions
Question 1
Solve $$x2=2 for $$x.
Enter each solution as a radical on the same line, separated by a comma.
Question 2
Solve $$x216−2=2 for $$x.
Enter each solution on the same line, separated by a comma.