We are tasked with solving the equation \( 5 + \sqrt{X} + 14 = X + 7 \) using deductive reasoning.

### Step-by-Step Deduction:

1. **Simplify the equation:**
   Start by simplifying both sides of the equation:
   \[
   5 + \sqrt{X} + 14 = X + 7
   \]
   Combine the constants on the left-hand side:
   \[
   19 + \sqrt{X} = X + 7
   \]

2. **Isolate the square root term:**
   To isolate \( \sqrt{X} \), subtract 7 from both sides:
   \[
   19 - 7 + \sqrt{X} = X
   \]
   Simplifying gives:
   \[
   12 + \sqrt{X} = X
   \]
   Subtract 12 from both sides to further isolate \( \sqrt{X} \):
   \[
   \sqrt{X} = X - 12
   \]

3. **Square both sides:**
   To remove the square root, square both sides of the equation:
   \[
   (\sqrt{X})^2 = (X - 12)^2
   \]
   This gives:
   \[
   X = (X - 12)^2
   \]

4. **Expand and simplify:**
   Now expand \( (X - 12)^2 \):
   \[
   X = X^2 - 24X + 144
   \]

5. **Rearrange the equation:**
   Bring all terms to one side to set the equation to zero:
   \[
   0 = X^2 - 25X + 144
   \]
   Or:
   \[
   X^2 - 25X + 144 = 0
   \]

6. **Solve the quadratic equation:**
   We will solve this using the quadratic formula:
   \[
   X = \frac{-(-25) \pm \sqrt{(-25)^2 - 4(1)(144)}}{2(1)}
   \]
   Simplifying:
   \[
   X = \frac{25 \pm \sqrt{625 - 576}}{2}
   \]
   \[
   X = \frac{25 \pm \sqrt{49}}{2}
   \]
   \[
   X = \frac{25 \pm 7}{2}
   \]
   So, the two possible solutions are:
   \[
   X = \frac{25 + 7}{2} = 16
   \]
   \[
   X = \frac{25 - 7}{2} = 9
   \]

### Final step: Check for extraneous solutions.
We need to substitute both values into the original equation to ensure they are valid solutions.

- For \( X = 16 \):
  \[
  5 + \sqrt{16} + 14 = 16 + 7
  \]
  \[
  5 + 4 + 14 = 23
  \]
  \[
  23 = 23 \quad \text{(True, so \( X = 16 \) is a solution)}
  \]

- For \( X = 9 \):
  \[
  5 + \sqrt{9} + 14 = 9 + 7
  \]
  \[
  5 + 3 + 14 = 16
  \]
  \[
  22 \neq 16 \quad \text{(False, so \( X = 9 \) is not a solution)}
  \]

### Conclusion:
The solution is \( X = 16 \).

---

Would you like more details or clarifications?

Here are 5 questions for further exploration:
1. How can the quadratic formula be used in different contexts?
2. What is an extraneous solution, and why does it arise?
3. How can we check for the validity of solutions in equations with square roots?
4. What are alternative methods for solving quadratic equations?
5. Can we solve this equation graphically?

**Tip:** When solving equations involving square roots, always check for extraneous solutions!

5 + √X + 14 = X + 7 use deductive reasoning

Master the process of solving the radical equation 5 + √X + 14 = X + 7 using deductive reasoning. This problem involves simplifying a square root equation and solving the resulting quadratic. Learn step-by-step how to apply the quadratic formula and check for extraneous solutions. Ideal for students in Grades 9-10.

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