Question

find the root of a quadratic equation by the method of a completing the square
4x²+4√3 x+3=0

Answer 1

$\underline{\mathfrak{ Solution : }}$

$\mathsf{ \implies 4{x}^{2} \: + \: 4\sqrt{3}x \: + \: 3 \: = \: 0} \\ \\ \textsf{ Divide both sides by 4, } \\ \\ \mathsf{ \implies \dfrac{4{x}^{2} \: + \: 4\sqrt{3}x \: + \: 3 \:}{4} = \:\dfrac{ 0}{4}}$

$\mathsf{\implies \dfrac{4{x}^{2}}{4} \: + \: \dfrac{4\sqrt{3}x}{4} \: + \: \dfrac{3}{4} \: = \: 0} \\ \\ \mathsf{ \implies {x}^{2} \: + \: \sqrt{3} x \: + \: \dfrac{3}{4} \: = \: 0 } \\ \\ \mathsf{ \implies {x}^{2} \: + \: 2 \: \cdot\: x \: \cdot \: \dfrac{ \sqrt{3} }{2} \: = \: \dfrac{ - 3}{4} }$

$\mathsf{ Add \: {(\dfrac{\sqrt{3}}{2})}^{2} \: to \: both \: sides, } \:$
$\mathsf{ \implies {x}^{2} \: + \: 2 \: \cdot\: x \: \cdot \: \dfrac{ \sqrt{3} }{2} \: + \: {(\dfrac{\sqrt{3}}{2})}^{2} = \: \dfrac{ - 3}{4} \: + \: {(\dfrac{\sqrt{3}}{2})}^{2} }$

$\textsf{ Using Algebraic identity : } \\ \\ \boxed{\mathsf{\implies {a}^{2} \: + \: 2ab \: + {b}^{2} \: = \: { ( a \: + \: b )}^{2} }}$

$\mathsf{ \implies { ( x \: + \: \dfrac{\sqrt{3}}{2} )}^{2} \: = \: \cancel{\dfrac{-3}{4}} \: + \: \cancel{ \dfrac{3}{4}} } \\ \\ \mathsf{ \implies{ (x \: + \: \dfrac{ \sqrt{3} }{2} )}^{2} \: = \: 0} \\ \\ \mathsf{ \implies( x \: + \: \dfrac{ \sqrt{3} }{2} )\: = \: \sqrt{0} } \\ \\ \mathsf{ \implies x \: + \: \dfrac{ \sqrt{3} }{2} \: = \: 0} \\ \\ \mathsf{ \therefore \: \: x \: = \: \dfrac{ - \sqrt{3} }{2} }$


$\boxed{\mathfrak{ Hope \: it \: helps !! }}$

Answer 2

Step-by-step explanation:

$Solution:</p><p></p><p>\begin{gathered}\mathsf{ \implies 4{x}^{2} \: + \: 4\sqrt{3}x \: + \: 3 \: = \: 0} < br / > \\ \\ \textsf{ Divide both sides by 4, } \\ \\ \mathsf{ \implies \dfrac{4{x}^{2} \: + \: 4\sqrt{3}x \: + \: 3 \:}{4} = \:\dfrac{ 0}{4}} \end{gathered}⟹4x2+43x+3=0  Divide both sides by 4, ⟹44x2+43x+3=40</p><p></p><p>\begin{gathered}\mathsf{\implies \dfrac{4{x}^{2}}{4} \: + \: \dfrac{4\sqrt{3}x}{4} \: + \: \dfrac{3}{4} \: = \: 0} \\ \\ \mathsf{ \implies {x}^{2} \: + \: \sqrt{3} x \: + \: \dfrac{3}{4} \: = \: 0 } \\ \\ \mathsf{ \implies {x}^{2} \: + \: 2 \: \cdot\: x \: \cdot \: \dfrac{ \sqrt{3} }{2} \: = \: \dfrac{ - 3}{4} }\end{gathered}⟹44x2+443x+43=0⟹x2+3x+43=0⟹x2+2⋅x⋅23=4−3</p><p></p><p>\mathsf{ Add \: {(\dfrac{\sqrt{3}}{2})}^{2} \: to \: both \: sides, } < br / > \:Add(23)2tobothsides, </p><p>\mathsf{ \implies {x}^{2} \: + \: 2 \: \cdot\: x \: \cdot \: \dfrac{ \sqrt{3} }{2} \: + \: {(\dfrac{\sqrt{3}}{2})}^{2} = \: \dfrac{ - 3}{4} \: + \: {(\dfrac{\sqrt{3}}{2})}^{2} }⟹x2+2⋅x⋅23+(23)2=4−3+(23)2</p><p></p><p>\begin{gathered}\textsf{ Using Algebraic identity : } \\ \\ < br / > < br / > \boxed{\mathsf{\implies {a}^{2} \: + \: 2ab \: + {b}^{2} \: = \: { ( a \: + \: b )}^{2} }} < br / > \end{gathered} Using Algebraic identity :  ⟹a2+2ab+b2=(a+b)2 </p><p></p><p>\begin{gathered}\mathsf{ \implies { ( x \: + \: \dfrac{\sqrt{3}}{2} )}^{2} \: = \: \cancel{\dfrac{-3}{4}} \: + \: \cancel{ \dfrac{3}{4}} } < br / > < br / > \\ \\ \mathsf{ \implies{ (x \: + \: \dfrac{ \sqrt{3} }{2} )}^{2} \: = \: 0} \\ \\ \mathsf{ \implies( x \: + \: \dfrac{ \sqrt{3} }{2} )\: = \: \sqrt{0} } \\ \\ \mathsf{ \implies x \: + \: \dfrac{ \sqrt{3} }{2} \: = \: 0} \\ \\ \mathsf{ \therefore \: \: x \: = \: \dfrac{ - \sqrt{3} }{2} }\end{gathered}⟹(x+23)2=4−3+43 ⟹(x+23)2=0⟹(x+23)=0⟹x+23=0∴x=2−3</p><p>$ slide your finger to see answer