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.

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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.

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