Solve for x: √(x + 3) = √(2x - 1)

Steps to Solve

1. Square both sides of the equation To eliminate the square roots, we square both sides of the given equation: $$ (\sqrt{x} + 3)^2 = (\sqrt{2x - 1})^2 $$

2. Expand both sides Expanding both sides of the equation gives us: $$ x + 6\sqrt{x} + 9 = 2x - 1 $$

3. Isolate the square root term Rearrange the equation to isolate the term with the square root: $$ 6\sqrt{x} = 2x - 1 - x - 9 $$ $$ 6\sqrt{x} = x - 10 $$

4. Square both sides again To eliminate the remaining square root, we square both sides of the equation again: $$ (6\sqrt{x})^2 = (x - 10)^2 $$ $$ 36x = x^2 - 20x + 100 $$

5. Rearrange into a quadratic equation Rearrange the equation into standard quadratic form: $$ x^2 - 20x - 36x + 100 = 0 $$ $$ x^2 - 56x + 100 = 0 $$

6. Solve the quadratic equation We can solve the quadratic equation using the quadratic formula: $$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$ Where $a = 1$, $b = -56$, and $c = 100$. Plugging in these values, we get: $$ x = \frac{56 \pm \sqrt{(-56)^2 - 4(1)(100)}}{2(1)} $$ $$ x = \frac{56 \pm \sqrt{3136 - 400}}{2} $$ $$ x = \frac{56 \pm \sqrt{2736}}{2} $$ $$ x = \frac{56 \pm \sqrt{16 \cdot 171}}{2} $$ $$ x = \frac{56 \pm 4\sqrt{171}}{2} $$ $$ x = 28 \pm 2\sqrt{171} $$

7. Find the possible values of x This gives us two possible solutions: $x_1 = 28 + 2\sqrt{171}$ and $x_2 = 28 - 2\sqrt{171}$

8. Approximate the values of x $$ \sqrt{171} \approx 13.07 $$ $$ x_1 \approx 28 + 2(13.07) \approx 28 + 26.14 \approx 54.14 $$ $$ x_2 \approx 28 - 2(13.07) \approx 28 - 26.14 \approx 1.86 $$

9. Check the solutions in the original equation Check $x_1 \approx 54.14$: $$ \sqrt{54.14} + 3 \approx 7.36 + 3 \approx 10.36 $$ $$ \sqrt{2(54.14) - 1} \approx \sqrt{108.28 - 1} \approx \sqrt{107.28} \approx 10.36 $$ Check $x_2 \approx 1.86$: $$ \sqrt{1.86} + 3 \approx 1.36 + 3 \approx 4.36 $$ $$ \sqrt{2(1.86) - 1} \approx \sqrt{3.72 - 1} \approx \sqrt{2.72} \approx 1.65 $$ Since $x_2$ does not satisfy the original equation, it is an extraneous solution.

10. Exact check of the solutions in the original equation Check $x = 28 + 2\sqrt{171}$: $$ \sqrt{28 + 2\sqrt{171}} + 3 = \sqrt{2(28 + 2\sqrt{171}) - 1} $$ $$ \sqrt{28 + 2\sqrt{171}} + 3 = \sqrt{56 + 4\sqrt{171} - 1} $$ $$ \sqrt{28 + 2\sqrt{171}} + 3 = \sqrt{55 + 4\sqrt{171}} $$ This solution checks out

Check $x = 28 - 2\sqrt{171}$: $$ \sqrt{28 - 2\sqrt{171}} + 3 = \sqrt{2(28 - 2\sqrt{171}) - 1} $$ $$ \sqrt{28 - 2\sqrt{171}} + 3 = \sqrt{56 - 4\sqrt{171} - 1} $$ $$ \sqrt{28 - 2\sqrt{171}} + 3 = \sqrt{55 - 4\sqrt{171}} $$ Now $\sqrt{28 - 2\sqrt{171}}$ is approximately equal to $1.36$ as shown above in our approximate check. So $\sqrt{55 - 4\sqrt{171}}$ is approximately equal to $1.65$ as shown above in our approximate check. Since $1.36 + 3 \neq 1.65$, we reject this solution because it is extraneous

11. Final Answer $x = 28 + 2\sqrt{171}$