Question

Find the roots of the following quadratic equation, if they exist, by the method of completing the square:

4x² + 4√3x + 3 = 0

Answer 1

Given: 4x² + 4√3x + 3 = 0 

⇒ 4x² + 4√3x = -3

⇒ (2x)² + 2(2x)√3 = -3 

LHS is of the form a² + 2ab where b = √3. 

∴ Adding b² = (√3)² = 3 on both sides, we get 

(2x)² + 2(2x)(√3) + (√3)² = -3 + (√3)² 

⇒ (2x + √3)² = -3 + 3 = 0 

∴ (2x + √3)² = 0 

⇒ 2x + √3 = 0 

⇒ 2x = -√3 

⇒ x = \(\frac{-\sqrt{3}}{2}\)

∴ The roots are \(\frac{-\sqrt{3}}{2}\) , \(\frac{-\sqrt{3}}{2}\) .