Question

Which constant should be added and subtracted to solve the quadratic equal \[4{x^2} - \sqrt {3}x - 5 = 0\] by the method of completing the square?

Answer 1

Hint: Here we are asked that which constant should be added and subtracted to solve the quadratic equal \[4{x^2} - \sqrt {3}x - 5 = 0\] by using method of completing the square. So firstly we will divide the whole equation by \[4\] and by the uses of the mathematical operations make it as the complete square with rearrangement. Later on we will solve it to get the required solution.

Complete step-by-step answer:
As we are given the quadratic equation as \[4{x^2} - \sqrt {3}x - 5 = 0\]
So now we need to divide the whole equation by \[4\] and we get the equation as
 \[{x^2} - \dfrac{{\sqrt 3 }}{4}x - \dfrac{5}{4} = 0\]
Therefore after rearrangement we get
 \[{\left( {x - \dfrac{{\sqrt 3 }}{8}} \right)^2} - \dfrac{5}{4} = \dfrac{3}{{64}}\]
Later on after simplification
 \[4{\left( {x - \dfrac{{\sqrt 3 }}{8}} \right)^2} - 5 - \dfrac{3}{{16}} = 0\]
Now we will compare the equation with general quadratic equation \[a{x^2} + bx + c = 0\] and will get the result as \[\dfrac{3}{{16}}\]
There to make \[4{x^2} - \sqrt {3}x - 5 = 0\] a complete square \[\dfrac{3}{{16}}\] must be added

Note: In the above question we have learned how to solve a problem by the use of complete square method. Remember the exact steps while solving such types of questions. The steps which we need to remember are the complete square method for solving the quadratic equation \[a{x^2} + bx + c = 0\] . Be careful with the signs while shifting the values of the one side of the equation to another side, that is we always need to change the sign from positive to negative and vice versa.