%5E2-b%5E2%3D0.jpeg "(x+a)^2-b^2=0")
Solving the Equation (x + a)^2 - b^2 = 0
The equation (x + a)^2 - b^2 = 0 is a quadratic equation in disguise. It can be solved using a few simple steps.
Understanding the Equation
The equation is based on the difference of squares factorization:
- a² - b² = (a + b)(a - b)
We can apply this to our equation:
(x + a)² - b² = [(x + a) + b][(x + a) - b] = 0
Solving for x
Now, we have a product of two factors that equals zero. This means at least one of the factors must be zero. Therefore, we have two possible solutions:
-
(x + a) + b = 0
- Solving for x, we get: x = -a - b
-
(x + a) - b = 0
- Solving for x, we get: x = -a + b
Example
Let's say we have the equation: (x + 3)² - 4 = 0
- We identify a = 3 and b = 2.
- Using the formulas from above:
- x = -3 - 2 = -5
- x = -3 + 2 = -1
Therefore, the solutions to the equation (x + 3)² - 4 = 0 are x = -5 and x = -1.
Conclusion
By recognizing the equation as a difference of squares, we can easily factor it and solve for x. This simple technique allows us to find the two solutions to the equation.