Plug in the coefficients into each box. If the equation has a 'missing term' (for example, no X^{2} term), then enter it as a zero.
x^{6} + x^{5} + x^{4} + x^{3} + x^{2} + x + = 0 |
x_{1}: + i |
x_{2}: + i |
x_{3}: + i |
x_{4}: + i |
x_{5}: + i |
x_{6}: + i |
Instructions: This calculator solves equations of the form Ax^{6} + Bx^{5} + Cx^{4} + Dx^{3} + Ex^{2} + Fx + G = 0, so if your equation is not in this form already, perform the necessary algebra to do so. Do not leave the field blank Due to round-off errors, some answers may be slightly off. |
This calculator solves for real and complex roots for a sextic equation in the form of Ax^{6} + Bx^{5} + Cx^{4} + Dx^{3} + Ex^{2} + Fx + G = 0 where A, B, C, D, E, F, and G are real numbers and A is non-zero. What this calculator does is computes the first root using Newton's Method (the initial guess is complex). If it converges with a complex root, it prints out its conjugate as a second root; if it converges to a real root, it divides out that root to get a quintic and then gets the second root by running Newton's Method on this quintic. After the two roots are found, it divides the two roots out to get a quartic and solves the quartic using the quartic formula implemented in our Quartic/Cubic/Quadratic Equation Calculator.
Error Codes:
1 - A non-numerical value has been detected in one of the fields. Once removed, you should not get this error.
2 - The constant term is zero, which means that x = 0 is a root to the entered equation. The other 5 roots can be found by factoring out x and plugging in the resulting quintic into the Quintic Equation Calculator. This calculator has had problems with equations that do not have a constant term, so the calculator does not allow.
3 - The Leading coefficient is zero, so the entered equation is not hexic; rather, the entered equation is a quintic (degree 5). Use the Quintic Equation Calculator to solve your equation.
4 - Division by zero occurred when the calculator attempted to compute the first solution. Pressing 'Calculate' again will give a new initial guess and should result in a solution. You may have to press the 'Calculate' again button multiple times. If you are still getting this error code after several attempts, shoot me an e-mail (you can find my e-mail address on the front page).
5 - Division by zero occurred when the calculator attempted to compute the second solution. Pressing 'Calculate' again will give a new initial guess and should result in a solution. You may have to press the 'Calculate' button again more than once. If you are still getting this error code after several attempts, shoot me an e-mail (you can find my e-mail address on the front page).
6 - The x^{5} coefficient is significantly higher than the others. There are a few precision issues with equations with a high x^{5} coefficent. After dividing the leading coefficient, the x^{5} coefficient must be less than 400 times the sum of the other coefficients.
7 - Newton's Method diverged while calculating the first solution or the 6,500 iteration limit has been exceeded. Pressing 'Calculate' again will result in a new random first guess and should result in a solution. You may have to press the 'Calculate' again button multiple times. If you are still getting this error code after several attempts, shoot me an e-mail (you can find my e-mail address on the front page).
8 - Newton's Method diverged while calculating the second solution or the 6,500 iteration limit has been exceeded. Pressing 'Calculate' again will result in a new random first guess and should result in a solution. You may have to press the 'Calculate' again button multiple times. If you are still getting this error code after several attempts, shoot me an e-mail (you can find my e-mail address on the front page).