We call equations like $x-7=2$x−7=2 linear equations. These are equations where all variables have a power of $1$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 $x=9$x=9
If instead, the variables in the equation have a power of $2$2, we call them quadratic equations. Quadratic equations can potentially have two solutions. For example, in the equation $x^2-7=2$x2−7=2, there are two solutions, $x=3$x=3 and $x=-3$x=−3.
Exploration
Solve $x^2+4=20$x2+4=20 for $x$x.
Following our rules for solving linear equations, we want to isolate $x$x and whatever we do to one side of the equation we do to the other. So our first step is to subtract $4$4 from both sides of the equation, giving us $x^2=16$x2=16.
Next we want to undo raising $x$x to the power of $2$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.
| $x^2$x2 | $=$= | $16$16 | |
| $x$x | $=$= | $\pm\sqrt{16}$±√16 |
Taking the positive and negative square root of both sides |
| $=$= | $\pm4$±4 |
Evaluating the positive and negative square roots |
The symbol $\pm$± 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 $x=4$x=4 and $x=-4$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$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 $\pm$± means "plus or minus".
If we can rearrange a quadratic equation into the form $x^2=k$x2=k, then we can solve the equation by taking the positive and negative square roots. That is, $x=\pm\sqrt{k}$x=±√k.
Practice questions
Question 1
Solve $x^2=2$x2=2 for $x$x.
Enter each solution as a radical on the same line, separated by a comma.
Question 2
Solve $\frac{x^2}{16}-2=2$x216−2=2 for $x$x.
Enter each solution on the same line, separated by a comma.