Question
Factorize:
$21\text{x}^2-2\text{x}+\frac{1}{21}$
$21\text{x}^2-2\text{x}+\frac{1}{21}$
✓
Answer
$21 x^2-2 x+\frac{1}{21}$
$=(\sqrt{21 x})^2-2 \sqrt{21} x \times \frac{1}{\sqrt{21}}+\left(\frac{1}{\sqrt{21}}\right)^2$
Using the identity $(x-y)^2=x^2+y^2-2 x y$
$\left(\sqrt{21} x-\frac{1}{\sqrt{21}}\right)^2$
$\therefore 21 x^2-2 x+\frac{1}{21}=\left(\sqrt{21} x-\frac{1}{\sqrt{21}}\right)^2$
$=(\sqrt{21 x})^2-2 \sqrt{21} x \times \frac{1}{\sqrt{21}}+\left(\frac{1}{\sqrt{21}}\right)^2$
Using the identity $(x-y)^2=x^2+y^2-2 x y$
$\left(\sqrt{21} x-\frac{1}{\sqrt{21}}\right)^2$
$\therefore 21 x^2-2 x+\frac{1}{21}=\left(\sqrt{21} x-\frac{1}{\sqrt{21}}\right)^2$
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