%5E2-25.jpeg "(x-4)^2-25")
Factoring and Solving (x-4)^2 - 25
The expression (x-4)^2 - 25 represents a quadratic equation in a slightly disguised form. Let's break down how to factor and solve it:
Recognizing the Difference of Squares
The key to simplifying this expression lies in recognizing the pattern of difference of squares:
- a^2 - b^2 = (a + b)(a - b)
In our case, we can rewrite (x-4)^2 as (x-4)(x-4). Therefore:
(x-4)^2 - 25 = (x-4)(x-4) - 5^2
Now we can clearly see the difference of squares pattern with a = (x-4) and b = 5.
Factoring the Expression
Applying the difference of squares formula:
(x-4)(x-4) - 5^2 = [(x-4) + 5][(x-4) - 5]
Simplifying the expressions inside the brackets:
= (x + 1)(x - 9)
Solving for x
To find the values of x that satisfy the equation (x-4)^2 - 25 = 0, we set the factored expression equal to zero:
(x + 1)(x - 9) = 0
This equation holds true when either factor is equal to zero:
- x + 1 = 0 => x = -1
- x - 9 = 0 => x = 9
Therefore, the solutions to the equation (x-4)^2 - 25 = 0 are x = -1 and x = 9.
Summary
We successfully factored and solved the expression (x-4)^2 - 25 by recognizing the difference of squares pattern. This method allows us to simplify the expression and find the solutions to the equation.