Q: Transcendental equation in radicals How can I solve this transcendental equation in radicals? $$\sqrt{2x^2-3x+4}=x$$ A: hint: Use the following identity $$\sqrt{u}=\frac{\sqrt{a^2+b^2}}{\sqrt{a^2-b^2}}$$ $$\sqrt{2x^2-3x+4}=\frac{\sqrt{2(2x-1)^2-3(2x-1)+4}}{\sqrt{(2x-1)^2-3(2x-1)+4}}$$ $$\sqrt{2x^2-3x+4}=\frac{2x-1}{\sqrt{(2x-1)^2-3(2x-1)+4}}$$ Determination of chemical composition of Lactobacillus delbrueckii ssp. bulgaricus and Streptococcus thermophilus subsp. thermophilus by ¹H and ¹³C NMR spectroscopy and multivariate analysis. Water-soluble extracts from the cell-free supernatant of two strains of Lactobacillus delbrueckii ssp. bulgaricus and Streptococcus thermophilus subsp. thermophilus were characterized by (1)H and (13)C NMR spectroscopy, to identify the polysaccharide fraction. In the (1)H NMR spectra of both the supernatants, signals attributable to alpha-D-glucose units were observed, assigned to the 1→4 linkages and to the terminal glucosyl residues. The signals arising from the D-glucopyranose residues of the alpha-1,3 glycosidic bonds were assigned. The (13)C NMR spectra of the supernatants, after the removal of the polysaccharide fraction, revealed signals attributable to carbohydrate and fatty acid components. To identify and quantify the polysaccharides and to investigate the relationships among the strains, an untargeted approach, based on the combination of chemometric tools (Principal Component Analysis, PCA, and Hierarchical Cluster Analysis, HCA), was used to explore the data. (1