Question

Solve $\sqrt{\frac{3}{2}(x-1)+\sqrt{2x^2-7x-4}}=$

Answer 1

$\begin{array}{c}\sqrt{\frac{3}{2}(x-1)+\sqrt{2x^2-7x-4}}\\ =\sqrt{\frac{3}{2}(x-1)+\sqrt{2x^2-8x+x-4}}\\ =\sqrt{\frac{3}{2}(x-1)+\sqrt{2x(x-4)+1(x-4)}}\\ =\sqrt{\frac{3x-3}{2}+\sqrt{(2x+1)(x-4)}}\\ =\sqrt{\frac{2x+x-4+1}{2}+2\times \frac{1}{2}\sqrt{(2x+1)(x-4)}}\\ =\sqrt{\frac{(2x+1)+(x-4)}{2}+2\times \frac{\sqrt{(2x+1)}}{\sqrt{2}}\times \frac{\sqrt{(x-4)}}{\sqrt{2}}}\\ =\sqrt{\frac{(2x+1)}{2}+\frac{(x-4)}{2}+2\times \frac{\sqrt{(2x+1)}}{\sqrt{2}}\times \frac{\sqrt{(x-4)}}{\sqrt{2}}}\\ =\sqrt{{\left(\frac{\sqrt{(2x+1)}}{\sqrt{2}}\right)}^2+{\left(\frac{\sqrt{(x-4)}}{\sqrt{2}}\right)}^2+2\times \frac{\sqrt{(2x+1)}}{\sqrt{2}}\times \frac{\sqrt{(x-4)}}{\sqrt{2}}}\\ =\sqrt{{\left(\frac{\sqrt{(2x+1)}}{\sqrt{2}}+\frac{\sqrt{(x-4)}}{\sqrt{2}}\right)}^2}=\frac{\sqrt{(2x+1)}}{\sqrt{2}}+\frac{\sqrt{(x-4)}}{\sqrt{2}}\\ =\frac{\sqrt{(2x+1)}+\sqrt{(x-4)}}{\sqrt{2}}.\end{array}$