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Solve the radical equation. Check all proposed solutions.
StartRoot 4 x plus 5 EndRoot equals x minus 44x+5=x−4
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Part 1
Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.
A.
The solution set is StartSet nothing EndSet .enter your response here.
​(Use a comma to separate answers as needed. Simplify your​ answer.)
B.
The solution set is the empty set.

It seems like the problem involves solving a radical equation of the form:

\[
\sqrt{4x + 5} = x - 4
\]

To solve this equation, we'll follow these steps:

### Step 1: Isolate the Square Root and Square Both Sides

Isolate the square root by squaring both sides of the equation:

\[
\left(\sqrt{4x + 5}\right)^2 = (x - 4)^2
\]

Simplifying this, we get:

\[
4x + 5 = (x - 4)^2
\]

### Step 2: Expand the Right Side

Expand the square on the right side:

\[
4x + 5 = x^2 - 8x + 16
\]

### Step 3: Rearrange into a Quadratic Equation

Bring all terms to one side to form a quadratic equation:

\[
x^2 - 8x - 4x + 16 - 5 = 0
\]

Simplify the terms:

\[
x^2 - 12x + 11 = 0
\]

### Step 4: Solve the Quadratic Equation

Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = -12 \), and \( c = 11 \).

\[
x = \frac{-(-12) \pm \sqrt{(-12)^2 - 4(1)(11)}}{2(1)}
\]

Simplify:

\[
x = \frac{12 \pm \sqrt{144 - 44}}{2}
\]

\[
x = \frac{12 \pm \sqrt{100}}{2}
\]

\[
x = \frac{12 \pm 10}{2}
\]

This gives two solutions:

\[
x = \frac{12 + 10}{2} = 11 \quad \text{and} \quad x = \frac{12 - 10}{2} = 1
\]

### Step 5: Check Both Solutions in the Original Equation

1. For \( x = 11 \):

   \[
   \sqrt{4(11) + 5} = \sqrt{44 + 5} = \sqrt{49} = 7
   \]

   And \( x - 4 = 11 - 4 = 7 \). Since both sides match, \( x = 11 \) is a solution.

2. For \( x = 1 \):

   \[
   \sqrt{4(1) + 5} = \sqrt{4 + 5} = \sqrt{9} = 3
   \]

   And \( x - 4 = 1 - 4 = -3 \). Since both sides do not match, \( x = 1 \) is not a valid solution.

### Final Answer

The solution set is:

\[
\text{The solution set is } \{11\}.
\]

Would you like further details or have any questions? Here are some related questions you might explore:

1. How do you solve radical equations involving different radicals?
2. What are the common mistakes when solving equations involving radicals?
3. How does squaring both sides affect the possible solutions of an equation?
4. Why is it important to check solutions when solving equations with radicals?
5. Can there be extraneous solutions in all radical equations?

**Tip:** Always check your proposed solutions back in the original equation, especially when dealing with radical expressions, as squaring can introduce extraneous solutions.

Learn how to solve the radical equation StartRoot 4x + 5 EndRoot = x - 4 with a detailed step-by-step solution. This problem involves radical equations and quadratic equations, requiring careful application of the quadratic formula. Suitable for high school students.

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