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:

- x² = 4
**[pure]** - 9x² – 5x = 0
**[incomplete]** - 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 2^{nd} and 1^{st} 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 1^{st}, 2^{nd}, and constant term, using the **quadratic formula **and electronically using a calculator. All complete quadratics follow the general form: y = **a**x² + **b**x + **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 = **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.

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 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:

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.