Question

How do you complete the square to solve $4{{x}^{2}}-6x+3=0$ ?

Answer 1

Hint: From the question we have been asked to find the solution to the equation $4{{x}^{2}}-6x+3=0$ using the completing square method. To solve this question first, we have to eliminate $4$ from the coefficient of ${{x}^{2}}$ and check whether its coefficient is \[1\]. Then we will proceed with our further calculation by adding an integer or fraction on both sides of the equation. so, we proceed with our solution as follows.

Complete step by step solution:
From the question given we have to solve the $4{{x}^{2}}-6x+3=0$ by using complete square method.
$\Rightarrow 4{{x}^{2}}-6x+3=0$
So, in the process of solving the question first we will remove the $4$ before ${{x}^{2}}$ by dividing $4$on both sides of the equation.
So, the equation will be reduced as follows.
$\Rightarrow {{x}^{2}}-\dfrac{6}{4}x+\dfrac{3}{4}=0$
$\Rightarrow {{x}^{2}}-\dfrac{3}{2}x+\dfrac{3}{4}=0$
Here we must make sure that the coefficient of ${{x}^{2}}$ should be 1 and nothing other than that, from the above equation we can clearly see that its coefficient is 1. So, we will further continue our solution as follows.
Here we will add and subtract the integer $\dfrac{9}{16}$ to the equation for the simplification. So, the equation will be reduced as follows.
$\Rightarrow {{x}^{2}}-\dfrac{3}{2}x+\dfrac{3}{4}+\dfrac{9}{16}-\dfrac{9}{16}=0$
$\Rightarrow {{x}^{2}}-\dfrac{3}{2}x+\dfrac{9}{16}+\dfrac{3}{16}=0$
Here now we will bring the integer $\dfrac{9}{16}$ to the right-hand side of the equation. So, the equation will be reduced as follows.
$\Rightarrow {{x}^{2}}-\dfrac{3}{2}x+\dfrac{9}{16}=-\dfrac{3}{16}$
Now, here we can clearly see that the expression on the left-hand side of the equation is a complete square. So, after rewriting the equation we the equation reduced as follows.
$\Rightarrow {{\left( x-\dfrac{3}{4} \right)}^{2}}=\dfrac{3}{16}$
Therefore, in this way we complete the square $\Rightarrow {{\left( x-\dfrac{3}{4} \right)}^{2}}=\dfrac{3}{16}$ to solve the question.
The solution for the equation will be as follows.
$\Rightarrow {{\left( x-\dfrac{3}{4} \right)}^{2}}=\dfrac{3}{16}$
Here we will shift the square of the left-hand side to the right-hand side then it will be square root to the right-hand side equation.
$\Rightarrow \left( x-\dfrac{3}{4} \right)=\sqrt{\dfrac{3}{16}}$
$\Rightarrow \left( x-\dfrac{3}{4} \right)=\pm \dfrac{\sqrt{3}}{4}$
$\Rightarrow x=\dfrac{3}{4}\pm \dfrac{\sqrt{3}}{4}$
Therefore, the solution for the question using complete square method will be $x=\dfrac{3}{4}\pm \dfrac{\sqrt{3}}{4}$ and in this way we complete the square $4{{x}^{2}}-6x+3=0$ to solve the question.

Note: Students should know the meaning and method of complete square method. Students while applying the complete square method should check whether the coefficient of ${{x}^{2}}$ should be 1. If it is not \[1\] then our process will not follow the complete square method. Hence, our solution will be wrong.