Lesson Introduction into Quadratic Equations

I am writing this lesson so that it contains everything you need to solve quadratic equations and do well on tests. Other quadratic lessons in this module explain the fine points of quadratics if you are interested. You can also try the quadratic equation solver that also shows you a graph.

What is a quadratic equation

A quadratic equation is an equation of form that involves only two things besides numbers: a variable and a square of this variable. Examples: , , and so on. Usually, they are arranged so that the square part goes first, then the part with the variable, and some constant, while the right side is equal to zero. In your tests, a, b and c will be actual numbers.

Solving Quadratic Equations

There are two ways of solving quadratics: factoring and using the Quadratic formula (see solver). Of these methods, the Quadratic Formula is the most reliable method that will give you the correct answer without guesswork. Here's how it works.

Let's say that you have an equation . If your equation is in some other form, for example , convert it to standard form with first, x part second, and the number third. The previous example , for example, converts to .

First, compute the discriminant. . You need to remember this formula. The discriminant can be positive, zero, or negative.


If the discriminant is positive, the quadratic equation has two different solutions (roots). One solution is . The other solution is . Click here to read the proof of these formulas. These solutions are so similar, that they are often written as . The +/- sign means that one solution comes with a "+" sign, and the other with a "-" sign. Last formula is called quadratic formula.


If the discriminant is equal to zero, the quadratic equation has one solution (root). This solution is . You can still use the same formula, but because the discriminant (which is an expression under the square root) is zero, it doesn't contribute, and +- doesn't make the difference. So, the root in this case is simply .


If the discriminant is negative, the quadratic equation has no solution (root) in the area of real numbers.


Example 1.
Solve quadratic equation by using the quadratic formula.
First calculate the discriminant: .
Discriminant is positive, so the equation has two roots:

and
.
The two solutions are 6 and -1

Example 2.
Solve quadratic equation by using the quadratic formula.
First calculate the discriminant: .
Discriminant is positive, so the equation has two roots:

and

The two solutions are and

Example 3.
Solve quadratic equation by using the quadratic formula.
Calculating the discriminant: .
Discriminant is equal to zero, so the equation has only one root:

The solution is 5.

Example 4.
Solve quadratic equation by using the quadratic formula.
Calculating the discriminant: .
Discriminant is negative, so the equation has no root in real numbers.
The equation has no solution in real numbers.