%5E2%3D(9-x)%5E2.jpeg "(x-7)^2=(9-x)^2")
Solving the Equation: (x-7)^2 = (9-x)^2
This equation involves squaring expressions, which can be a bit tricky to handle. However, there's a clever trick we can use to simplify it.
Understanding the Problem
The equation (x-7)^2 = (9-x)^2 implies that the expressions inside the parentheses are either equal or their values have the same absolute value (but opposite signs).
Let's break it down:
- Squaring: Both sides of the equation are squared, meaning they are multiplied by themselves.
- Equality: The equation states that the results of the squaring are equal.
Solving the Equation
Here's how we can solve the equation:
- Take the square root of both sides: This gets rid of the squares and simplifies the equation.
√((x-7)^2) = √((9-x)^2) - Simplify: The square root of a squared expression is the expression itself.
(x-7) = (9-x) - Solve for x:
x + x = 9 + 7 2x = 16 x = 8
Conclusion
Therefore, the solution to the equation (x-7)^2 = (9-x)^2 is x = 8.
Important Note: We should be aware that taking the square root can introduce extraneous solutions. To ensure our solution is valid, we should always plug it back into the original equation and verify that it holds true. In this case, substituting x = 8 back into the original equation confirms that it's the correct solution.