Question

find the root of the following quadratic equation, if they exist by the method of completing the squares:
(1) 4x2 + 4√3x + 3 = 0​

Answer 1

Step-by-step explanation:

Nature of roots is determined by the discriminant(value under square root).   In general, discriminant = b² - 4ac,   for ax² + bx + c = 0.

Here,

a = 4    ;    b = 4√3    ;  c = 3

 Thus, discriminant is,

⇒ b² - 4ac

⇒ (4√3)² - 4(4)(3)

⇒ 48 - 48

⇒ 0,        it means, roots exist and are real & equal.

So,  now we can factorize.

⇒ 4x² + 4√3x + 3 = 0

⇒ x² + √3x + (3/4) = 0

⇒ x² + √3x = - 3/4

⇒ x² + 2(√3 /2)x + (√3 /2)² = -3/4 + (√3 /2)²

⇒ (x + √3 /2)² = -3/4 + 3/4

⇒ (x + √3 /2)² = 0

⇒ x + √3 /2 = 0

⇒ x = - √3/2