%5E2-5(x%2B2)%2B6.jpeg "(x+2)^2-5(x+2)+6")
Factoring Quadratic Expressions: (x+2)^2 - 5(x+2) + 6
This article will explore the factorization of the quadratic expression (x+2)^2 - 5(x+2) + 6. We will employ a technique known as substitution to simplify the expression and make factorization easier.
Understanding the Expression
The given expression, (x+2)^2 - 5(x+2) + 6, is a quadratic expression with a slightly complex structure. It involves the term (x+2) repeated multiple times. This suggests a potential simplification using substitution.
Applying Substitution
Let's introduce a new variable, say 'y', to represent (x+2).
y = x + 2
Substituting 'y' into the original expression, we get:
y^2 - 5y + 6
Factoring the Simplified Expression
Now we have a simple quadratic expression in terms of 'y'. We can factorize this expression by finding two numbers that multiply to give 6 and add up to -5. These numbers are -2 and -3.
Therefore, the factored form of y^2 - 5y + 6 is:
(y - 2)(y - 3)
Re-substituting to Get the Final Factorization
Finally, we substitute back 'x+2' for 'y' to get the final factored form of the original expression:
(x + 2 - 2)(x + 2 - 3)
Simplifying this, we get:
(x)(x - 1)
Conclusion
Therefore, the factored form of the quadratic expression (x+2)^2 - 5(x+2) + 6 is x(x - 1). By using substitution, we transformed a complex quadratic into a simpler form, making the factorization process straightforward.