The equation provided is:

$\sqrt{x + 2} + \sqrt{7x + 2} = 6$

Solution:

Step 1: Isolate one of the square roots. $\sqrt{x + 2} = 6 - \sqrt{7x + 2}$

Step 2: Square both sides to eliminate the square root on the left. $x + 2 = (6 - \sqrt{7x + 2})^2$ Expanding the right-hand side: $x + 2 = 36 - 12\sqrt{7x + 2} + (7x + 2)$ Simplifying: $x + 2 = 38 + 7x - 12\sqrt{7x + 2}$ Step 3: Isolate the remaining square root. $x + 2 - 38 - 7x = -12\sqrt{7x + 2}$ $-6x - 36 = -12\sqrt{7x + 2}$ Divide by -12 on both sides: $\frac{6x + 36}{12} = \sqrt{7x + 2}$ $\frac{x}{2} + 3 = \sqrt{7x + 2}$

Step 4: Square both sides again. $\left(\frac{x}{2} + 3\right)^2 = 7x + 2$ Expanding the left-hand side: $\frac{x^2}{4} + 3x + 9 = 7x + 2$ Step 5: Rearrange the equation. $\frac{x^2}{4} + 3x + 9 - 7x - 2 = 0$ Simplify: $\frac{x^2}{4} - 4x + 7 = 0$ Multiply through by 4 to clear the fraction: $x^2 - 16x + 28 = 0$

Step 6: Solve the quadratic equation. Using the quadratic formula: $x = \frac{-(-16) \pm \sqrt{(-16)^2 - 4(1)(28)}}{2(1)}$ $x = \frac{16 \pm \sqrt{256 - 112}}{2}$ $x = \frac{16 \pm \sqrt{144}}{2}$ $x = \frac{16 \pm 12}{2}$ So, the two possible solutions are: $x = \frac{16 + 12}{2} = 14 \quad \text{or} \quad x = \frac{16 - 12}{2} = 2$

Step 7: Check for extraneous solutions.

• For $x = 14$: $\sqrt{14 + 2} + \sqrt{7(14) + 2} = \sqrt{16} + \sqrt{100} = 4 + 10 = 14 \quad (\text{False, so discard this solution})$
• For $x = 2$: $\sqrt{2 + 2} + \sqrt{7(2) + 2} = \sqrt{4} + \sqrt{16} = 2 + 4 = 6 \quad (\text{True})$

Thus, the solution is: $x = 2$

Would you like further details on any step?

Follow-up Questions:

1. What are extraneous solutions and why do they occur?
2. Can this method be generalized for other square root equations?
3. What would happen if the equation had a cubic root instead of a square root?
4. How do you solve quadratic equations with complex roots?
5. Why is squaring both sides of an equation sometimes problematic?

Tip: Always check for extraneous solutions when dealing with square roots, as squaring both sides can introduce false solutions.