1.

Solve the quadratic equations by factorization method only:x2 + x + 1 = 0

Answer»

Given as x2 + x + 1 = 0

x2 + x + 1/4 + 3/4 = 0

x2 + 2 (x) (1/2) + (1/2)2 + 3/4 = 0

(x + 1/2)2 + 3/4 = 0 [Since, (a + b)2 = a2 + 2ab + b2]

(x + 1/2)2 + 3/4 × 1 = 0

As we know, i2 = –1 ⇒ 1 = –i2

On substituting 1 = –i2 in the above equation, we get

(x + 1/2)2 + 3/4 (-1)2 = 0

(x + 1/2)2 + 3/4 i2 = 0

(x + 1/2)2 – (3i/2)2 = 0

[On using the formula, a2 – b2 = (a + b) (a – b)]

(x + 1/2 + 3i/2) (x + 1/2 – 3i/2) = 0

(x + 1/2 + 3i/2) = 0 or (x + 1/2 – 3i/2) = 0

x = -1/2 – 3i/2 or x = -1/2 + 3i/2

∴ The roots of the given equation are -1/2 + 3i/2, -1/2 – 3i/2



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