Question

simplify (root3+2)² ​

Answer 1

Answer:

$( \sqrt{3 } + 2) {}^{2} \\ = ( \sqrt{3}) {}^{2} + (2) {}^{2} + 2 \times 2 \times \sqrt{3 } \\ = 3 + 4 + 4 \sqrt{3 } \\ = 7 + 4 \sqrt{3}$

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Answer 2

Question :-

Simplify : $\mathtt {(\sqrt 3 + 2)^2}$

$\begin {gathered} \end {gathered}$

Solution :-

Putting $\bold {\sqrt 3 = a}$ and $\bold{2 = b}$ , we have -

$\begin {gathered} \end {gathered}$

$\begin {aligned} \bold {(\sqrt 3 + 2)^2} &=\boxed {\mathtt{(a + b)^2 = a^2+2ab+b^2}}\\\\&\Rightarrow (\sqrt 3)^2 + 2 \times \sqrt 3 \times 2 + 2^2\\\\&\Rightarrow 3 + 4\sqrt3 + 4\ \ \boxed {\because \bold {(\sqrt a)^2 = a \implies (\sqrt 3)^2 = 3}}\\\\\therefore \bold {(\sqrt 3 + 2)^2} &= \underline {\boxed {\bf 7 + 4\sqrt3}}\ \star \end {aligned}$

$\begin {gathered} \end {gathered}$

Additional Information :-

Some more Formulae used for Factorization :-

$\bullet \mathtt{(a - b)^2 = a^2 - 2ab + b^2}$

$\bullet\ \mathtt{a^2 - b^2 = (a + b)(a-b)}$

$\bullet\ \mathtt{(x+a)(x+b) = x^2+(a+b)\:x+ab}$

$\bullet\ \mathtt{(x+y+z)^2 = x^2+y^2+z^2+2xy+2yz+2xz}$

$\begin {aligned} \bullet\ \mathtt{(x+y)^3} &= \mathtt{x^3+y^3+3xy\:(x+y)}\\\\ &\rightarrow \mathtt {x^3 + y^3 + 3x^2y + 3xy^2} \end {aligned}$

$\begin {aligned} \bullet\ \mathtt{(x-y)^3} &= \mathtt{x^3-y^3-3xy\:(x-y)}\\\\ &\rightarrow \mathtt {x^3 - y^3 - 3x^2y + 3xy^2} \end {aligned}$

$\bullet \ \mathtt {x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx)}$