# How to Solve an Equation in GeoGebra

You can solve equations either symbolically or numerically. `GeoGebra` solves them automatically when you enter them in `CAS` and click `Symbolic Evaluation` or `Numeric Evaluation`. `Symbolic Evaluation` gives you the exact solution, while `Numeric Evaluation` gives you the solution as a possibly approximated decimal number.

If you get $\left\{\right\}$ as the output, then the equation has no solution. If you get $\left\{x=a\right\}$, the equation has one solution, and if you get $\left\{x=a,x=b\right\}$, the equation has two solutions. Note! Here $a$ and $b$ represent numbers.

`GeoGebra` Instruction 1

### Solutionin`CAS`

1.
Open `CAS` under  `View` in  `Menu`.
2.
Enter the equation in the `CAS` window. If the right-hand side of the equation is 0, you only need to type the expression on the left-hand side.
3.
Click `Solve`  in the `Menubar` to get the exact solution, or click `Solve Numerically`  in the `Menubar` to solve numerically.

Solving an equation graphically is the same as drawing the graph of the expressions on each side of the equation, and finding the $x$-coordinate of the intersection between the graphs.

`GeoGebra` Instruction 2

### Solutionin`Graphics View`

1.
Choose `Algebra View` and `Graphics View` in the  `View` menu.
2.
Enter the expressions into two separate rows in the `Algebra View` and press `Enter`. Give the expressions fitting names.
3.
Find the intersection between the graphs. You do this by typing

Intersect(<object>, <object>)

where the objects correspond to the two expressions. Press `Enter`.

4.
If the equation has any solutions, you can find them in the `Algebra View`.

Example 1

Solve the equation

 $f\left(x\right)=g\left(x\right)$

graphically, where

$\begin{array}{llll}\hfill f\left(x\right)& ={x}^{2}-2x-5\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill g\left(x\right)& =2{x}^{2}+x-15\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

The picture below shows the solutions to the equation that you get by following the instructions above.