Question

Find the roots of the following quadratic equation (if they exist) by the method of completing the square.
\[4{{x}^{2}}+4\sqrt{3}x+3=0\]

Answer 1

Hint: In algebra, a quadratic equation (from the Latin quadratus for "square") is any equation that can be rearranged in standard form as \[a{{x}^{2}}+bx+c=0\] where x represents an unknown, and a, b, and c represent known numbers, where a ≠ 0. If a = 0, then the equation is linear, not quadratic, as there is no \[a{{x}^{2}}\] term. The numbers a, b, and c are the coefficients of the equation and may be distinguished by calling them, respectively, the quadratic coefficient, the linear coefficient and the constant or free term.

Complete step-by-step solution -
Now, for solving a quadratic equation, we are asked to use the completing the square method.
In this method, we make the quadratic into a perfect square and then equate it to the value that has been added to the quadratic equation to make it into a perfect square. For this method, the coefficient of the leading term of the quadratic equation should be 1.
As mentioned in the question, we have to find the roots of the given quadratic equation by using the method of completing the square.
Now, we will follow the procedure of completing the square to get the roots of the given quadratic equation as follows
\[\begin{align}
  & 4{{x}^{2}}+4\sqrt{3}x+3=0 \\
 & {{x}^{2}}+\sqrt{3}x+\dfrac{3}{4}=0 \\
 & {{x}^{2}}+\sqrt{3}x+\dfrac{3}{4}=0 \\
 & {{x}^{2}}+2\dfrac{\sqrt{3}}{2}x+{{\left( \dfrac{\sqrt{3}}{2} \right)}^{2}}=0 \\
 & {{\left( x+\dfrac{\sqrt{3}}{2} \right)}^{2}}=0 \\
\end{align}\]
Now, on taking square root of both the sides, we get
\[\begin{align}
  & x+\dfrac{\sqrt{3}}{2}=0 \\
 & x=-\dfrac{\sqrt{3}}{2} \\
\end{align}\]
Hence, the given or asked quadratic equation has only one root and that is as follows
\[x=-\dfrac{\sqrt{3}}{2}\]

NOTE: -
The students can make an error while solving this question if they don’t know what a quadratic equation is and the method of completing the square to get the roots of the equation.
Also, for solving a quadratic equation, the students can use the quadratic formula to get to the roots.