- 34 lessons
- 0 quizzes
- 7 week duration
Unit 1: Linear Systems
Unit 2: Analytic Geometry
Unit 3: Geometric Properties
Unit 4: Quadratic Relations
Most relations that you have studied in mathematics have been linear. However, many non-linear also exist in real life.
Unit 5: Quadratic Expressions
Unit 6: Quadratic Equations
Unit 7: Trigonometry
Factor a Difference of Squares
So far we’ve learned three factoring techniques. The first one, common factoring, is a technique that can be used for any polynomial. The other two were specific for quadratics, namely trial-and-error and decomposition.
The whole purpose behind factoring any quadratic – if you haven’t discovered already – is convert it in such a format that enables you to solve for the x’s (the roots). Otherwise, if it’s in general form, you can’t easily do that.
You’ll find out here that some quadratics whose b-term is missing and whose a and c terms are being subtracted, can be factored by another technique known the difference of squares. Examples of quadratics that fit this pattern look like this:
- y = x² – 100 (easy)
- y = 98a² – 450b² (medium)
- y = (3x + 8)² – (x – 2)² (hard)
Also, as the name of the technique implies, both terms in the quadratic need to be perfect squares (squarootable).
Take equation (1) as an example. Taking the square root of x² yields x. In other words, √x² = x. Similarly, √100 = 10. In the second equation, after common factoring 2 from both terms, you get:
= 2(49a² – 225b²)
Notice now that √49a² = 7a and √225b² = 15b. Both terms within the parentheses are perfect squares; hence, this technique can be used.
In equation (3), the square root of (3x + 8)² is 3x + 8 and the square root of (x – 2)² is x – 2; therefore it fits the same mold. A summary of what’s required is written below.
Conditions required to factor a difference of squares:
- b term needs to be missing.
- The two remaining terms are being subtracted.
- The two remaining terms need to be a perfect square.
Let’s what a few examples: