In algebra we must take into account some properties that simplify time and the correct way to develop a problem. For this reason, we present to you the topic of difference of squares , highly developed in the chapter on notable products.
With the theory and examples that we will provide you, there will be no doubts about the subject. Take note!
Difference of squares
The difference of squares refers to 2 algebraic terms of the form:
a² – b²
The result of this difference is:
a² – b² = (a-b)(a+b)
It should be noted that this difference of squares results in a product of 2 factors, one as a difference and the other as a sum; that is, the expression is factored by applying said property ad.
Examples:
1Develop: x² – 1²
Applying the learned property we have:
x² – 1² = (x – 1)(x + 1)
2Develop: m² – n²
⇒ m² – n² = (m – n)(m + n)
3Factor: 3² – a²
⇒ 3² – a² = (3 – a)(3 + a)
4Factor: 4x² – 9
In this example it is not in difference of squares form, but we can do the following:
⇒ 4x² – 9 = (2x)² – 3²
So, we have:
(2x)² – 3² = (2x – 3)(2x + 3); luego:
∴ 4x² – 9 = (2x – 3)(2x + 3)
5Factor: 64 – b²
⇒ 64 – b² = 8² – b² = (8 – b)(8 + b)
6Factor: 9x² – 25²
⇒ 9x² – 25² = (3x)² – 5² = (3x – 5)(3x + 5)
7Factor: y² – 900²
⇒ y² – 900² = y² – 30² = (y – 30)(y + 30)
8Factor: 9x 4 – 4y²
⇒ 9x 4 – 4y² = (3x²)² – (2y)² = (3x² – 2y)(3x² + 2y)
9Resolver: 16x8 – 1
⇒ 16x8 – 1 = (4x4)² – 1² = (4x4 – 1)(4x4 + 1)
Observation:
We can also write the difference of squares property in reverse, like this:
(a-b)(a+b) = a² – b²
Let’s see an application example of this form.
10Calcular: M = 2020² – 2022.2018
To develop this example, we will use some tricks from the statement, note:
⇒ M = 2020² – 2022.2018 = 2020² – (2020 + 2)(2020 – 2)
Applying difference of squares, we have:
⇒ M = 2020² – (2020² – 2²)
Developing:
∴ M = 4
