The most basic method uses these steps:

- Simplify both sides (if necessary).
- If there are any
*variable*terms on the right-hand side, then subtract these terms from both sides (and simplify). - If there are no variable terms left in the equation, then you have a statement that is always true or always false, and that is your final answer; otherwise continue below.
- If there is a constant term on the left-hand side, then subtract this term from both sides (and simplify).
- If there is now a coefficient (or just a minus sign)
on the
*leading*term on the left-hand side, then divide both sides by that coefficient (or −1) (and simplify). - If the left-hand side is linear, then you should have the answer now; otherwise continue below.
- If there is more than one term on the left-hand side,
then add a constant to both sides
that makes the left-hand side into a perfect square;
this step is called
*completing the square*. - If there is more than one term on the left-hand side, then factor the left-hand side (while you simplify the right-hand side).
- Take square roots of both sides (and simplify with ± on the right-hand side).
- If there is now a constant term on the left-hand side, then subtract this term from both sides (and simplify).

Alternatively, you can use this method:

- Simplify both sides (if necessary).
- If there are any terms
*at all*on the right-hand side, then subtract these terms from both sides (and simplify). - If the left-hand side is linear,
then solve the equation using a method for linear equations;
otherwise use the
*quadratic formula*below.

*a**x*^{2}+*b**x*+*c*= 0, and*a*≠ 0;*x*= [−*b*± √(*b*^{2}− 4*a**c*)]/(2*a*).

Finally, you can try to solve the equation by factoring. That's usually the fastest way when it works, but it doesn't always work; the other two methods are guaranteed to work every time.

Go back to the course homepage.

This web page was written between 2007 and 2017 by Toby Bartels, last edited on 2017 November 8. Toby reserves no legal rights to it.

The permanent URI of this web page
is
`http://tobybartels.name/MATH-1100/2020FA/equations/`

.