Question
Factorize:
$2\text{a}^2+2\sqrt{6}\text{ab}+3\text{b}^2$
$2\text{a}^2+2\sqrt{6}\text{ab}+3\text{b}^2$
✓
Answer
$2\text{a}^2+2\sqrt{6}\text{ab}+3\text{b}^2$
$=\big(\sqrt{2}\text{a}\big)^2+2\times\sqrt{2}\text{a}\times\sqrt{3}\text{b}+\big(\sqrt{3}\text{b}\big)^2$
Using the identity $(p+q)^2=p^2+q^2+2 p q$
$=\big(\sqrt{2}\text{a}+\sqrt{3}\text{b}\big)^2$
$=\big(\sqrt{2}\text{a}+\sqrt{3}\text{b}\big)\big(\sqrt{2}\text{a}+\sqrt{3}\text{b}\big)$
$\therefore2\text{a}^2+2\sqrt{6}\text{ab}+3\text{b}^2$
$=\big(\sqrt{2}\text{a}+\sqrt{3}\text{b}\big)\big(\sqrt{2}\text{a}+\sqrt{3}\text{b}\big)$
$=\big(\sqrt{2}\text{a}\big)^2+2\times\sqrt{2}\text{a}\times\sqrt{3}\text{b}+\big(\sqrt{3}\text{b}\big)^2$
Using the identity $(p+q)^2=p^2+q^2+2 p q$
$=\big(\sqrt{2}\text{a}+\sqrt{3}\text{b}\big)^2$
$=\big(\sqrt{2}\text{a}+\sqrt{3}\text{b}\big)\big(\sqrt{2}\text{a}+\sqrt{3}\text{b}\big)$
$\therefore2\text{a}^2+2\sqrt{6}\text{ab}+3\text{b}^2$
$=\big(\sqrt{2}\text{a}+\sqrt{3}\text{b}\big)\big(\sqrt{2}\text{a}+\sqrt{3}\text{b}\big)$
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