Previous page : Complex roots
Next page : Analysis of the Complex Quadratic equation
Complex Quadratic equations
On this page you can explore quadratic equations interactively. You can change the values of a, b and c in az2+bz+c=0 by moving the corresponding points (in red, green and blue). They are spaced a bit above the real axis to separate them slightly, but they are, initially, all real numbers. The roots are shown as violets points in an Argand diagram. The corresponding real quadratic function y=ax2+bx+c is also shown, but now the vertical axis is a real axis corresponding to y rather than an imaginary axis.
You can also see grey circles at fixed distanced from the origin.
You can move a turquoise point corresponding to z, to calculate the value of f(z)=az2+bz+c.
You can do more, but first I have suggestions about things to explore.
- Move a so that the solutions lie on a one of the grey circles, say with the radius two. Now change the value of b and notice how the roots move. Try to figure out why. More on this on the following page.
- Now try the same with a and c.
- You can click on the button on the top after “parameters are”. When in complex setting the values of the parameters, a,b, and c can be freely moved in the Argand diagram.
The next thing you might test is to turn on a coloured field. Depending on if the setting you may look at the real or imaginary part of f(z)=az2+bz+c. Red through yellowish means larger and larger positive values. Dark blue to light blue more and more negative values. Black means close to zero. You can also look at the modulus (the absolute value) of the function.
- Switch between the real and imaginary field. How are they compared to each other?
- Set the parameters to complex and move around a in circles around the origin. Why are the field rotating in the other direction?
- With the real field the zeroes are at the vertices of something that looks like dark parabolas, and with the complex field a bit into the parabolas. Why so?
- You may also turn on a direction field where you can see the argument (angle) of the reult of the function.
- You may also change between standard form and vertex form, and in the other direction too. If any of the parameters would force the corresponding points to be outside the diagram you will get a warning. Just check if any of the parameters get outside the range –5 to 5.
So, time to explore.
Up a level : Complex numbersPrevious page : Complex roots
Next page : Analysis of the Complex Quadratic equation
Last modified: