%5E2-4(a-3b)-21.jpeg "(a-3b)^2-4(a-3b)-21")
Factoring the Expression: (a-3b)^2 - 4(a-3b) - 21
This expression can be factored using a few steps, making it easier to work with in various mathematical contexts.
Step 1: Recognizing a Pattern
Observe that the expression has a repeated term: (a-3b). This suggests we can use substitution to simplify the expression.
Step 2: Substitution
Let's replace (a-3b) with a single variable, say x. Our expression now becomes:
x² - 4x - 21
Step 3: Factoring the Quadratic
This is a simple quadratic expression that can be factored by finding two numbers that multiply to -21 and add up to -4. These numbers are -7 and 3.
Therefore, the factored form of the quadratic is:
(x - 7)(x + 3)
Step 4: Substituting Back
Now, let's substitute (a-3b) back in for x:
(a-3b - 7)(a - 3b + 3)
Final Factored Expression
The fully factored form of the original expression is:
(a-3b - 7)(a - 3b + 3)
This is the simplest form of the expression, and it can be useful for solving equations, simplifying expressions, or performing other mathematical operations.