Find the roots of the following quadratic (if they exist) by the method of completing the square. 4x^2 + 4√3x + 3 = 0

Question

Find the roots of the following quadratic (if they exist) by the method of completing the square.

\(4x^2+4\sqrt{3}x+3=0\)

Answer 1

Given: The quadratic equation \(4x^2+4\sqrt{3}x+3=0\)

To find: the roots of the following quadratic (if they exist) by the method of completing the square.

Solution: We have to make the quadratic equation a perfect square if possible or sum of perfect square with a constant.

Step 1: Make the coefficient of x² unity.In the equation \(4x^2+4\sqrt{3}x+3=0,\) The coefficient of x² is 4.So to make the coffecient of x² equals to 1.divide the whole equation by 4.

The quadratic equation now becomes:

Step 2: Shift the constant term on RHS,

\(\Rightarrow x^2+\sqrt{3}x=-\frac{3}{4}\)

Step 3: Add square of half of coefficient of x on both the sides.

Step 4: Apply the formula,(a + b)² = a² + 2ab + b² on LHS and solve RHS,

Here a = x and b \(=\frac{\sqrt{3}{}}{2}\)

Step 5: As the RHS is zero, the roots exist.Since the quadratic equations have 2 roots, in this case both roots will be same.