%5E2-49%3D0.jpeg "(x-1)^2-49=0")
Solving the Equation (x-1)² - 49 = 0
This article explores the solution of the quadratic equation (x-1)² - 49 = 0. We will utilize the difference of squares factorization to solve for the values of x.
Understanding the Difference of Squares
The difference of squares pattern is a fundamental concept in algebra. It states that:
a² - b² = (a + b)(a - b)
Applying this pattern to our equation:
(x - 1)² - 49 = 0
We can rewrite the equation as:
(x - 1)² - 7² = 0
Now, by applying the difference of squares formula, we get:
[(x - 1) + 7][(x - 1) - 7] = 0
Simplifying the expression:
(x + 6)(x - 8) = 0
Finding the Solutions
For the product of two factors to be zero, at least one of the factors must be equal to zero. Therefore, we have two possible solutions:
-
x + 6 = 0
- x = -6
-
x - 8 = 0
- x = 8
Conclusion
Hence, the solutions to the equation (x-1)² - 49 = 0 are x = -6 and x = 8.
This problem demonstrates the power of recognizing algebraic patterns and applying them to simplify and solve equations efficiently.