Solving Second Degree Equations by the Quadratics Formula

A second-degree equation (also known as a quadratic equation) is one whose highest-degree term is of second degree. Generally, the highest degree in any equation dictates the maximum possible number of solutions. Thus a quadratic equation, being of degree 2, has up to two solutions or roots. This is the same reason why all the first-degree equations we solved in the previous lesson always had a single solution.

Solving quadratic equations is definitely an art form. There are many things you need to look out for before devising a plan to solve. The first thing you need to determine is whether the quadratic is pure, incomplete, or complete. Examples of each type is shown below:

  1. x² = 4   [pure]
  2. 9x² – 5x = 0   [incomplete]
  3. 4x² – 5x + 2 = 0   [complete]

Pure quadratics are the easiest to solve. They contain an x² term and a constant only; no first-degree term exists. To solve pure quadratics, you isolate for x similar to the way you solve first-degree equations. Incomplete quadratics possess a 2nd and 1st degree term – solving these require a special technique called factoring that’ll be introduced in a later unit. Our main focus in this lesson is to solve complete quadratics, which contain a 1st, 2nd, and constant term, using the quadratic formula and electronically using a calculator. All complete quadratics follow the general form: y = ax² + bx + c. Take example 3, for instance, 4 presents a, –5 presents b, and +2 represents c.

The formula looks like this:

… and can be used to solve basically any quadratic equation, including pure and incomplete quadratics. 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.

Unlike first-degree equations which form a straight line when graphed, quadratics form a parabola (∪ or ∩). Therefore, depending on the equation, not all parabolas pass through the x-axis (left). 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:

Finally, a major part of this section is your ability to use technology (your calculator) to solve quadratics rather than waste time doing it manually using the formula. The video below explains how to use your Casio fx-991ES Plus to find the roots of any quadratic.