The Quadratic Formula

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 = ax² + bx + 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 roots

The discriminant is the part of the quadratic formula underneath 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: