2-%3D-4%E2%88%9A(x-4)-2-answer.jpeg "(x-1)2 = 4√(x-4) 2 Answer")
Solving the Equation: (x-1)^2 = 4√(x-4)
This equation involves a square root and a squared term, making it a bit more complex than a simple linear equation. Here's how we can solve it step-by-step:
1. Isolate the Square Root
First, we need to isolate the square root term on one side of the equation. To do this, we can divide both sides by 4:
(x-1)^2 / 4 = √(x-4)
2. Square Both Sides
To eliminate the square root, we square both sides of the equation:
[(x-1)^2 / 4]^2 = [√(x-4)]^2
This simplifies to:
(x-1)^4 / 16 = x-4
3. Expand and Rearrange
Expand the left side of the equation and then move all terms to one side:
(x^4 - 4x^3 + 6x^2 - 4x + 1) / 16 = x - 4
x^4 - 4x^3 + 6x^2 - 4x + 1 = 16x - 64
x^4 - 4x^3 + 6x^2 - 20x + 65 = 0
4. Solving the Quartic Equation
We now have a quartic equation (an equation with the highest power of x being 4). Unfortunately, there's no general formula for solving quartic equations like there is for quadratic equations. However, we can try to factor the equation or use numerical methods (like graphing calculators or software) to find the solutions.
In this specific case, the equation doesn't factor easily. We can use numerical methods to find the approximate solutions.
Solutions
Using numerical methods, we find that the equation has two real solutions:
- x ≈ 1.76
- x ≈ 4.24
Checking the Solutions
It's always a good idea to check our solutions by plugging them back into the original equation. Doing so will verify that both solutions satisfy the equation.
Important Note: When solving equations with square roots, it's important to be aware of extraneous solutions. These are solutions that we find during the solving process but don't actually satisfy the original equation. Always check your solutions to avoid this issue.