The standard form of the quadratic equation is a polynomial of the second degree, written as ,
where is unknown and are real numbers with . The equation is not quadratic
if , because the term is reduced to zero leaving only , which is the
equation of a line.
The Quadratic Formula
The easiest way to solve a quadratic equation is probably the quadratic formula. The quadratic formula can be derived by completing the
square on the quadratic equation . The quadratic formula is:
Using the quadratic formula to solve a quadratic equation is straight-forward, as shown by the following example:
Given the equation , we get:
Solution by Factoring
Solving quadratic equations can be done using various methods. One way to solve a quadratic equation is by factoring. It is
important to keep in mind, however, that not every quadratic equation can be factored. Factoring a trinomial expression is part
of another topic (please refer to Factoring Polynomials for a detailed step-by-step
Once the quadratic equation is factored, the roots of the equation can be easily found. For example, the quadratic expression
factors to . Since assigning a value of either to
satisifies the equation , the solution set is .
A slightly more complicated example is one such as . After factoring, it becomes .
We then need to solve each of the roots algebraically:
The solution set is therefore .
Completing the Square
Since it's not possible to factor every quadratic expression, we often solve quadratic equations by the method known as completing
the square. In order to solve a quadratic equation by completing the square, we perform the following steps:
1) Write the equation on the standard form .
2) Divide both sides of the equation by the coefficient of if not (i.e, divide by .)
3) Subtract the constant term from both sides (i.e, subtract .)
4) Divide the coefficient of by , square the result, then add the resulting value to both sides (i.e, add or .)
5) Factor the left side of the equation.
6) Apply the square root property
7) Check the results if required. (i.e, plug the results into the equation to test it.)
8) Write the solution set.
For example, given the quadratic equation , we follow the steps given above:
1) Write the equation in standard form:
2) Divide both sides of the equation by the coefficient of , which is :
3) Subtract the constant term from both sides:
4) Divide the coefficient of by , square the result, then add the resulting value to both sides:
5) Factor the left side of the equation:
6) Apply the square root property:
7) Check the results if required:
8) Write the solution set:
Since completing the square is more complex than factoring or using the quadratic formula, it's always a good idea to try to use those
methods before using this one (that is, unless you're doing homework and the assignment is to solve the problem by completing the square!)