Question

Factorise the following:
x⁴ $-$ 8x²y² + 12y⁴

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have}{x}^4-8x^2{y}^2+12y^4. \\ \mathrm{Let}{x}^2=a\mathrm{and}{y}^2=b \\ \mathrm{i}.\mathrm{e}.,x^4=a^2,y^4=b^2\mathrm{and}{x}^2{y}^2=ab \\ \therefore x^4-8x^2{y}^2+12y^4=a^2-8ab+12b^2 \\ =a^2-6ab-2ab+12b^2 \\ =a\left(a-6b\right)-2b\left(a-6b\right) \\ =\left(a-6b\right)\left(a-2b\right) \\ \mathrm{Resubstituting}\mathrm{the}\mathrm{values}\mathrm{of}a\mathrm{and}b,\mathrm{we}\mathrm{get}: \\ {\mathrm{x}}^4-8{\mathrm{x}}^2{\mathrm{y}}^2+12{\mathrm{y}}^4=\left({\mathrm{x}}^2-6{\mathrm{y}}^2\right)\left({\mathrm{x}}^2-2{\mathrm{y}}^2\right) \end{gathered}$$