The completely factored form of 8x² − 50 is 2(2x − 5)(2x + 5). The expression is first rewritten by removing its greatest common factor, and the remaining binomial is then factored using the difference of two squares pattern.
Introduction
The expression 8x² − 50 is a quadratic expression consisting of two terms. To factor it completely, both the numerical coefficients and the variable terms must be considered.
A common first attempt is to factor out 2:
8x² − 50 = 2(4x² − 25)
This is a correct step, but it is not the completely factored form. The expression inside the parentheses, 4x² − 25, can still be factored because it is a difference of two squares The details matter here..
The final result is:
2(2x − 5)(2x + 5)
What Does “Completely Factored Form” Mean?
An algebraic expression is in completely factored form when it is written as a product of factors that cannot be factored further using integer coefficients Practical, not theoretical..
Take this: 2(4x² − 25) is not completely factored because 4x² − 25 can be rewritten as:
(2x − 5)(2x + 5)
The factors 2x − 5 and 2x + 5 cannot be separated further while keeping integer coefficients. Therefore:
2(2x − 5)(2x + 5) is the completely factored form Which is the point..
Step-by-Step Factoring of 8x² − 50
Step 1: Identify the Greatest Common Factor
The expression is:
8x² − 50
The coefficients are 8 and 50. Their greatest common factor is 2:
- Factors of 8: 1, 2, 4, 8
- Factors of 50: 1, 2, 5, 10, 25, 50
The largest common factor is 2.
Divide each term by 2:
8x² ÷ 2 = 4x²
50 ÷ 2 = 25
Therefore:
8x² − 50 = 2(4x² − 25)
Step 2: Recognize the Difference of Two Squares
The expression inside the parentheses is:
4x² − 25
A difference of two squares has the form:
a² − b² = (a − b)(a + b)
Here, both terms are perfect squares:
4x² = (2x)²
25 = 5²
So:
4x² − 25 = (2x)² − 5²
Using the difference of squares formula:
(2x)² − 5² = (2x − 5)(2x + 5)
Step 3: Combine All Factors
Substitute the factored form of 4x² − 25 back into the expression:
2(4x² − 25)
becomes
2(2x − 5)(2x + 5)
Thus, the completely factored form is:
2(2x − 5)(2x + 5)
Scientific Explanation of the Factoring Process
Factoring is based on reversing multiplication. Instead of expanding
