Solving quadratic equations

A quadratic equation is an equation whose sides are both polynomials of degree at most 2. You can also say that an equation is quadratic in a particular variable if it would become a quadratic equation upon replacing all of the other variables with specific constants (no matter which constants you chose). A quadratic equation in one variable can always be solved, or more generally an equation that is quadratic in a variable may be solved for that variable, over the complex numbers (and therefore over the real numbers by throwing out imaginary solutions). There are two main methods for doing this.

The most basic method uses these steps:

At this point, you should have the answer.

Alternatively, you can use this method:

The quadratic formula says that the following two statements are equivalent: (Notice that a ≠ 0 precisely when the equation is nonlinear.) This formula may be derived by applying the first set of steps to the generic equation ax2 + bx + c = 0. The quadratic formula can introduce rounding errors when a is close to zero, and you might also be in a situation where you don't know whether a is zero or not; for alternative versions of the quadratic formula that can be applied as long as at least one of a, b, and c is nonzero (and which tends to avoid rounding errors as long as at least one of them is far from zero), see The numerical analyst's quadratic formula (optional, on an external site with fancy formatting possibilities).

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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