[tex] \huge \boxed{\mathfrak{Question} \downarrow}[/tex]
- Expand & simplify ⇨ [tex]( \sqrt{10} - \sqrt{2} ) ^{2} [/tex]. Give your answer in the form [tex]b - c \: \sqrt{5} [/tex] where b & c are integers.
[tex] \large \boxed{\mathbb{ANSWER\: WITH\: EXPLANATION} \downarrow}[/tex]
[tex]( \sqrt { 10 } - \sqrt { 2 } ) ^ { 2 }[/tex]
Use binomial theorem [tex]\left(a-b\right)^{2}=a^{2}-2ab+b^{2} [/tex] to expand [tex]\left(\sqrt{10}-\sqrt{2}\right)^{2}[/tex].
[tex]\left(\sqrt{10}\right)^{2}-2\sqrt{10}\sqrt{2}+\left(\sqrt{2}\right)^{2} [/tex]
The square of [tex]\sqrt{10}[/tex] is 10.
[tex]10-2\sqrt{10}\sqrt{2}+\left(\sqrt{2}\right)^{2} [/tex]
Factor [tex]10=2\times 5[/tex]. Rewrite the square root of the product [tex]\sqrt{2\times 5} [/tex] as the product of square roots [tex]\sqrt{2}\sqrt{5}[/tex].
[tex]10-2\sqrt{2}\sqrt{5}\sqrt{2}+\left(\sqrt{2}\right)^{2} [/tex]
Multiply [tex]\sqrt{2}[/tex] and [tex]\sqrt{2} [/tex] to get 2.
[tex]10-2\times 2\sqrt{5}+\left(\sqrt{2}\right)^{2} [/tex]
Multiply -2 and 2 to get -4.
[tex]10-4\sqrt{5}+\left(\sqrt{2}\right)^{2} [/tex]
The square of [tex]\sqrt{2}[/tex] is 2.
[tex]10-4\sqrt{5}+2 [/tex]
Add 10 and 2 to get 12.
[tex] \boxed{ \boxed{\bf\:12-4\sqrt{5} }}[/tex]
- Here, b & c are integers where [tex]\boxed{ \sf \: b = 12 \: and \: c = 4}[/tex]
