Question

Solve: \[3{{x}^{2}}-4\sqrt{3}x+4=0\]
A.The roots are \[\dfrac{1}{\sqrt{3}}\] and \[\dfrac{2}{\sqrt{5}}\]
B.The roots are \[\dfrac{1}{\sqrt{5}}\] and \[\dfrac{2}{\sqrt{3}}\]
C.The roots are \[\dfrac{7}{\sqrt{3}}\] and \[\dfrac{6}{\sqrt{5}}\]
D.None of these

Answer 1

Hint: To find the roots of the quadratic equation, we use the quadratic formula on the question, by using the long method formula:
The roots of the quadratic equation is \[\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\]
where \[a,\text{ }b,\text{ }c\] are referred from the equation written in the form of \[a{{x}^{2}}+bx+c=0\] .

Complete step-by-step answer:
To find the value we first write the value of the variable by forming the equation in form of \[a{{x}^{2}}+bx+c=0\] which is equivalent to \[3{{x}^{2}}-4\sqrt{3}x+4=0\] .
The values of the variables are written in the form of \[a=3\] , \[b=-4\sqrt{3}\] and \[c=4\] .
Placing the values in the formula of the quadratic equation we get the value of the roots as:
 \[\Rightarrow \dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\]
 \[\Rightarrow \dfrac{-\left( -4\sqrt{3} \right)\pm \sqrt{{{\left( 4\sqrt{3} \right)}^{2}}-4\times 3\times +4}}{2\times 3}\]
Simplifying the given equation, we get:
 \[\Rightarrow \dfrac{-\left( -4\sqrt{3} \right)\pm \sqrt{{{\left( 4\sqrt{3} \right)}^{2}}-4\times 3\times +4}}{2\times 3}\]
 \[\Rightarrow \dfrac{4\sqrt{3}\pm \sqrt{16\times 3-48}}{6}\]
 \[\Rightarrow \dfrac{4\sqrt{3}}{6}\pm \dfrac{\sqrt{16\times 3-48}}{6}\]
The term inside the root becomes zero as 48-48 cancels out,
 \[\Rightarrow \dfrac{2}{\sqrt{3}},\dfrac{2}{\sqrt{3}}\]
So, the correct answer is “Option D”.

Note: Another method to solve the question is by dividing the middle section into two equal parts like:
 \[\Rightarrow 3{{x}^{2}}-4\sqrt{3}x+4\]
 \[\Rightarrow 3{{x}^{2}}-2\sqrt{3}x-2\sqrt{3}x+4\]
 \[\Rightarrow \sqrt{3}x(\sqrt{3}x-2)-2\left( \sqrt{3}x-2 \right)=0\]
 \[\Rightarrow (\sqrt{3}x-2)\left( \sqrt{3}x-2 \right)=0\]
 \[\Rightarrow x=\dfrac{2}{\sqrt{3}},\dfrac{2}{\sqrt{3}}\]