Math, asked by samirmaths, 1 year ago

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

Answers

Answered by Anonymous
0

\underline{\mathfrak{ Solution : }}

\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}}

\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, }<br /> \:
 \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 : } \\ \\<br /><br />\boxed{\mathsf{\implies {a}^{2} \: + \: 2ab \: + {b}^{2} \: = \: { ( a \: + \: b )}^{2} }}<br />

\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} }


\boxed{\mathfrak{ Hope  \: it \:  helps !! }}
Answered by Anonymous
1

Step-by-step explanation:

Solution:</p><p></p><p>\begin{gathered}\mathsf{ \implies 4{x}^{2} \: + \: 4\sqrt{3}x \: + \: 3 \: = \: 0} &lt; br / &gt; \\ \\ \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&lt;br/&gt; 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, } &lt; br / &gt; \:Add(23)2tobothsides,&lt;br/&gt;</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 : } \\ \\ &lt; br / &gt; &lt; br / &gt; \boxed{\mathsf{\implies {a}^{2} \: + \: 2ab \: + {b}^{2} \: = \: { ( a \: + \: b )}^{2} }} &lt; br / &gt; \end{gathered} Using Algebraic identity : &lt;br/&gt;&lt;br/&gt;⟹a2+2ab+b2=(a+b)2&lt;br/&gt;</p><p></p><p>\begin{gathered}\mathsf{ \implies { ( x \: + \: \dfrac{\sqrt{3}}{2} )}^{2} \: = \: \cancel{\dfrac{-3}{4}} \: + \: \cancel{ \dfrac{3}{4}} } &lt; br / &gt; &lt; br / &gt; \\ \\ \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&lt;br/&gt;&lt;br/&gt;⟹(x+23)2=0⟹(x+23)=0⟹x+23=0∴x=2−3</p><p> slide your finger to see answer

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