Of the several methods we have for solving quadratics, the most useful is the **quadratic formula**. It will work for any quadratic, regardless of the type of roots, and it can easily be programmed into your calculator. Interestingly, the quadratic formula is derived by **completing the square** of the general form quadratic, y = ax² + bx + c.

- You’re likely
**not**expected to*derive*the quadratic formula from scratch, but in case you are, follow this link.

The formula looks like this:

… and can be used to solve basically any quadratic equation. Notice that the formula contains the same letters found in the general form, y = **a**x² + **b**x + **c**. So to use it correctly, you have to identify the **a**, **b**, and **c** coefficients in a quadratic trinomial and substitute them into the formula. Let’s see this in action:

A slightly more complicated question is provided underneath. Notice that quadratics with non-integer coefficients are ideal for the quadratic formula.

It’s also important to remember that not all quadratics pass through the x-axis; if that’s the case, the quadratic formula will give you an error when calculating the square-root part. In other words, sometimes given the way a quadratic is positioned on an x-y plane, it will not yield any solutions. We consider these roots as **non-real roots**. For example, try using the quadratic formula for the equation:

x² + 2x + 5 = 0

You’ll notice that the **radicand** (the number under a radical symbol) is -16. Any time your radicand is less than zero, your parabola doesn’t cross the x-axis (shown in figure).

Guidelines to determine the number of rootsThe discriminant is the part of the

quadratic formulaunderneath the square root symbol: b² − 4ac. The discriminant tells us whether there are two solutions, one solution, or no solutions.If b²−4ac = 0 (1 real root)

If b²−4ac > 0 (2 real root)

If b²−4ac < 0 (0 real root)

More on this is explained in the video below: