If you do not have a calculator that does combinations or permutations, there is another option to do those calculation. Desmos is know for its ability to graph formulas…but did you know it can also do calculations like a scientific calculator?
To find this scientific calculator, go to the Desmos website (www.desmos.com).
Located in the bottom left corner of the screen is a button to access the scientific calculator.Pressing this button takes you to the calculator. To access the command for permutations, press the button for func highlighted below.
This will take you to a new set of buttons. Select the button nPr shown below.
Pressing this button will paste the permutation command to the line above the buttons.
If you are computing 6P4 (or P(6,4)), you need to use your keyboard to type 6,4).
You can also do combinations in a similar manner.
If you type the carriage return on the keypad (dark blue button on the bottom right), it will go to a line on which you can put a new command.
To do a factorial like 10!, type in the number 10 on the keyboard.
Then select the ! button.
Notice that there are also other buttons that might prove useful like a square root, nth root, natural logarithm, and common logarithm buttons.
In a previous Math-FAQ, we looked at the different parts of a parabola. Based on this information, you know that to find the x intercepts of a parabola we need to solve a quadratic equation. When we solve a quadratic equation to find the x intercepts of the graph, you might expect to always have solutions. But as the Math-FAQ below shows, this is not always the case.
Students are often surprised when they graph a parabola a notice that the parabola has no x intercepts.
But as the graph above shows, parabolas do exist that do not cross the x axis.
However, suppose you do not have the graph of y = x2 +2x+3 available. How could you use the equation to determine whether this parabola has any x intercepts?
Let’s start by following the usual process for finding x intercepts of any graph. Set y = 0 to get the equation
This is a quadratic equation with a = 1, b = 2, and c = 3. To solve this equation, we need to use the quadratic formula:
Now put in the values for a, b, and c.
This gives us
This might set off alarms in your mathematical brain!
How can you take the square root of a negative number? For numbers graphed on a real number graph, you can’t. That is why our graph above has no x intercepts. However, if we expand our knowledge of numbers to complex numbers, we can write out a solution to the quadratic equation .
In complex numbers, is defined to be equal to the letter i. To evaluate the square root above, think of it as
Since i = and , we can simplify our square root as
our solution to the quadratic equation as
In short, the quadratic equation has a solution that
uses i. Since our graph does not allow for this type of number, it shows
no x intercepts.