(x-2)^2-9

Jasonbradley

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Jun 17, 2024

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Factoring and Solving the Expression (x-2)^2 - 9

The expression (x-2)^2 - 9 represents a quadratic equation. Let's explore how to factor it and find its solutions:

Factoring the Expression

We can factor this expression using the "difference of squares" pattern:

• Difference of Squares: a² - b² = (a + b)(a - b)

1. Recognize the Pattern: Notice that (x-2)² is a perfect square, and 9 is also a perfect square (3²).

2. Apply the Pattern: We can rewrite the expression as: (x-2)² - 9 = [(x-2) + 3][(x-2) - 3]

3. Simplify:
   [(x-2) + 3][(x-2) - 3] = (x + 1)(x - 5)

Therefore, the factored form of the expression is (x + 1)(x - 5).

Solving for x

To find the values of x that make the expression equal to zero, we set the factored form equal to zero and solve:

(x + 1)(x - 5) = 0

This equation is true if either factor is equal to zero:

• x + 1 = 0 => x = -1
• x - 5 = 0 => x = 5

Therefore, the solutions to the equation (x-2)² - 9 = 0 are x = -1 and x = 5.

Conclusion

By understanding the difference of squares pattern, we can easily factor the expression (x-2)² - 9 and find its solutions. This knowledge is fundamental in solving quadratic equations and simplifying algebraic expressions.