Solve the radical equation. Check all proposed solutions.

StartRoot x EndRootxplus+StartRoot x minus 24 EndRootx−24equals=4

We are given the equation:

\[
\sqrt{x} + \sqrt{x - 24} = 4
\]

### Step 1: Isolate one of the square roots
Let's start by isolating one of the square root terms. Subtract \(\sqrt{x - 24}\) from both sides:

\[
\sqrt{x} = 4 - \sqrt{x - 24}
\]

### Step 2: Square both sides
Next, square both sides of the equation to eliminate the square root on the left-hand side:

\[
(\sqrt{x})^2 = (4 - \sqrt{x - 24})^2
\]

Simplifying both sides:

\[
x = (4 - \sqrt{x - 24})^2
\]

Expanding the right-hand side using the formula \((a - b)^2 = a^2 - 2ab + b^2\):

\[
x = 16 - 8\sqrt{x - 24} + (x - 24)
\]

### Step 3: Simplify the equation
Now, simplify the equation:

\[
x = 16 - 8\sqrt{x - 24} + x - 24
\]

Combine like terms:

\[
x = x - 8 - 8\sqrt{x - 24}
\]

Subtract \(x\) from both sides:

\[
0 = -8 - 8\sqrt{x - 24}
\]

### Step 4: Isolate the square root
Add 8 to both sides:

\[
8 = -8\sqrt{x - 24}
\]

Now, divide both sides by \(-8\):

\[
-1 = \sqrt{x - 24}
\]

### Step 5: Solve for \(x\)
At this point, we square both sides:

\[
(-1)^2 = (\sqrt{x - 24})^2
\]

\[
1 = x - 24
\]

Solving for \(x\):

\[
x = 25
\]

### Step 6: Check the solution
We need to check if \(x = 25\) is a valid solution by substituting it back into the original equation:

\[
\sqrt{25} + \sqrt{25 - 24} = 4
\]

This simplifies to:

\[
5 + 1 = 4
\]

This is **false**, so \(x = 25\) is **not** a solution. Thus, the equation has no real solutions.

### Summary:
There are **no real solutions** to the given radical equation.

---

Would you like further details, or do you have any questions?

Here are 5 related questions to explore:
1. How do you solve a radical equation with more than one square root?
2. What are common pitfalls when squaring both sides of an equation?
3. How do you handle extraneous solutions in radical equations?
4. Why do we square both sides of the equation when solving radical equations?
5. Can a radical equation have no real solutions?

**Tip:** Always check your solutions when solving radical equations, as squaring both sides can introduce extraneous solutions.

Solve the radical equation. Check all proposed solutions. StartRoot x EndRootxplus+StartRoot x minus 24 EndRootx−24equals=4

Learn how to solve the radical equation √x + √(x - 24) = 4 with a detailed step-by-step solution. This problem involves isolating square roots, squaring both sides, and checking for extraneous solutions. Ideal for high school students in Grades 9-12.

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