Although a specific quadratic equation was used to introduce the method Solution by Completing the Square, the approach used there actually makes it possible for us to treat a whole class of equations ax² + bx + c = 0 by means of literal coefficients. In this section, we will derive the formula that we can use to solve the entire class of quadratic equations.

First, divide both sides by the coefficient a (a0, because a quadratic equation is assumed):

Next, add the right expression that can help transfer the first two terms into a square form to both sides:

Rearranging the terms, we have:

Combining the first three terms into a square, we have:

Now, move the right term to the left side:

Making the second main expression into a square:

Simplifying the second major term, we obtain:

Changing the equation into factored form, we get:

Therefore, the two solutions are:

Normally, we write two solutions together, as follows:

Discriminant

The expression under the radical symbol of the quadratic formula, b² - 4ac, is called a discriminant. As shown by the formula, the equation has no real solution when the discriminant is less than zero because the square root of a negative number does not exist within the real number system. When the discriminant equals zero, there is only one rational solution, -b/(2a). If the discriminant is bigger than zero, there are two real solutions. The above is a pure algebraic explanation.

Explained geometrically, the above three situations correspond to situations where:

1. the graph of the quadratic equation does not cross the x-axis;
2. it crosses the x-axis at one point;
3. it crosses the x-axis at two points.