To solve a quadratic equation means to find values for x that make the y side of the equation equal to zero. Recall that a quadratic has a highest degree of 2. 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. The two roots can be equal, different, or they may be imaginary or complex numbers (these terms will make more sense to you in our next lesson).
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.
Pure quadratics look like this (↓) and are the easiest to solve. They contain an x² term and a constant only; no first-degree term exists. For example:
- 3x² – 75 = y
- x² = 4
To solve these, you set y = 0 and find what your x values are. Given that quadratics can have up to 2 solutions, square rooting a number should be represented with ±. Let’s solve equation (1):
- Set y = 0 → 3x² – 75 = 0
- Shift the -75 to the right side → 3x² = +75
- Divide both sides by 3 → x² = 25
- Square root both sides → x = ±5
- Your solutions are: (5, 0) and (-5, 0)
An incomplete quadratic is one that has the second (x²) and first (x) degree term present, but not the constant term, c. Now, if you have a quadratic that’s not pure, but incomplete, you solve the same way except you have to common factor along the way. For example:
- y = x² + 5x
- Set y = 0 → x² – 5x = 0
- Make sure both terms are on one side to common factor. The common factor is x:
- x(x – 5) = 0 (Do not divide both sides by x!)
- Set each factor equal to 0:
- x = 0
- x – 5 = 0 → x = +5
- Your solutions are: (0, 0) and (5, 0)
A more thorough discussion on pure and incomplete quadratics is shown in the video below. In Part 2 of the lesson, we’ll look exclusively at how to solve complete quadratics – an introduction to complete quadratics is mentioned in the video as well. These consist of equations that are trinomials and with a first and second degree term, and a constant.