(x-2)^1/2-(28-2x)^1/4=0

3 min read Jun 17, 2024

Solving the Equation: (x-2)^1/2 - (28-2x)^1/4 = 0

This equation involves fractional exponents, which can be intimidating. However, we can solve it systematically using a combination of algebraic manipulation and substitution. Here's how we can approach it:

1. Isolating the Fractional Exponents

First, we want to isolate the terms with fractional exponents on one side of the equation. We can do this by adding the term with the 1/4 exponent to both sides:

(x-2)^1/2 = (28-2x)^1/4

2. Raising Both Sides to a Suitable Power

To eliminate the fractional exponents, we need to raise both sides of the equation to a power that will cancel them out. Since the least common multiple of 2 and 4 is 4, we'll raise both sides to the power of 4:

[(x-2)^1/2]^4 = [(28-2x)^1/4]^4

Simplifying, we get:

(x-2)^2 = (28-2x)

3. Expanding and Rearranging

Now we have a quadratic equation. Let's expand and rearrange it to the standard form:

x^2 - 4x + 4 = 28 - 2x x^2 - 2x - 24 = 0

4. Solving the Quadratic Equation

This quadratic equation can be solved by factoring:

(x - 6)(x + 4) = 0

Therefore, the possible solutions are:

x = 6 or x = -4

5. Checking for Extraneous Solutions

Since the original equation involved fractional exponents, it's essential to check if these solutions actually satisfy the original equation. This is because raising both sides to a power can introduce extraneous solutions.

Checking x = 6:

(6-2)^1/2 - (28-2*6)^1/4 = 4^1/2 - 16^1/4 = 2 - 2 = 0

This solution is valid.

Checking x = -4:

(-4-2)^1/2 - (28-2*-4)^1/4 = (-6)^1/2 - 36^1/4

This solution results in a square root of a negative number, which is not a real number. Therefore, x = -4 is an extraneous solution.

Conclusion

The only valid solution to the equation (x-2)^1/2 - (28-2x)^1/4 = 0 is x = 6.