sqrt(5x-4)-sqrt(x)=2
\(\begin{array}{|rcll|} \hline \sqrt{5x-4}-\sqrt{x} &=& 2 \quad &| \quad +\sqrt{x} \\ \sqrt{5x-4} &=& 2 +\sqrt{x} \quad &| \quad \text{square both sides} \\ 5x-4 &=& (2 +\sqrt{x})^2 \\ 5x-4 &=& 4+4\cdot \sqrt{x} + x \quad &| \quad -x \\ 4x-4 &=& 4+4\cdot \sqrt{x} \quad &| \quad :4 \\ x-1 &=& 1+ \sqrt{x} \quad &| \quad -1\\ x-2 &=& \sqrt{x} \quad &| \quad \text{square both sides} \\ (x-2)^2 &=& x\\ x^2-4x+4 &=& x \quad &| \quad -x \\ x^2-5x+4 &=& 0\\\\ x &=& \frac{5\pm \sqrt{25-4\cdot 4} } {2} \\ x &=& \frac{5\pm \sqrt{25-16} } {2} \\ x &=& \frac{5\pm \sqrt{9} } {2} \\ x &=& \frac{5\pm 3 } {2} \\\\ x_1 &=& \frac{5 + 3 } {2} \\ x_1 &=& \frac82 \\ \mathbf{x_1} & \mathbf{=} & \mathbf{4} \\\\ x_2 &=& \frac{5 - 3 } {2} \\ x_2 &=& \frac22 \\ \mathbf{x_2} & \mathbf{=} & \mathbf{1} \quad \text{ no solution}\\ \hline \end{array}\)
sqrt(5x-4)-sqrt(x)=2
\(\begin{array}{|rcll|} \hline \sqrt{5x-4}-\sqrt{x} &=& 2 \quad &| \quad +\sqrt{x} \\ \sqrt{5x-4} &=& 2 +\sqrt{x} \quad &| \quad \text{square both sides} \\ 5x-4 &=& (2 +\sqrt{x})^2 \\ 5x-4 &=& 4+4\cdot \sqrt{x} + x \quad &| \quad -x \\ 4x-4 &=& 4+4\cdot \sqrt{x} \quad &| \quad :4 \\ x-1 &=& 1+ \sqrt{x} \quad &| \quad -1\\ x-2 &=& \sqrt{x} \quad &| \quad \text{square both sides} \\ (x-2)^2 &=& x\\ x^2-4x+4 &=& x \quad &| \quad -x \\ x^2-5x+4 &=& 0\\\\ x &=& \frac{5\pm \sqrt{25-4\cdot 4} } {2} \\ x &=& \frac{5\pm \sqrt{25-16} } {2} \\ x &=& \frac{5\pm \sqrt{9} } {2} \\ x &=& \frac{5\pm 3 } {2} \\\\ x_1 &=& \frac{5 + 3 } {2} \\ x_1 &=& \frac82 \\ \mathbf{x_1} & \mathbf{=} & \mathbf{4} \\\\ x_2 &=& \frac{5 - 3 } {2} \\ x_2 &=& \frac22 \\ \mathbf{x_2} & \mathbf{=} & \mathbf{1} \quad \text{ no solution}\\ \hline \end{array}\)