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

Asked by Sakshi | 1 year ago |  270

##### Solution :-

Given: x2 + x + 1 = 0

$$x^2 + x + \dfrac{1}{4} + \dfrac{3}{4} = 0$$

x2 + 2 (x) ($$\dfrac{1}{2}$$) + ($$\dfrac{1}{2}$$)2 + $$\dfrac{3}{4}$$ = 0

(x + $$\dfrac{1}{2}$$)2 + $$\dfrac{3}{4}$$ = 0 [Since, (a + b)2 = a2 + 2ab + b2]

(x + $$\dfrac{1}{2}$$)2 + $$\dfrac{3}{4}$$ × 1 = 0

We know, i2 = –1 = 1 = –i2

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

(x + $$\dfrac{1}{2}$$)2 + $$\dfrac{3}{4}$$ (-1)2 = 0

(x + $$\dfrac{1}{2}$$)2 +  $$\dfrac{3}{4}$$i2 = 0

(x + $$\dfrac{1}{2}$$)2 – $$\sqrt{(\dfrac{3i}{2})^2}=0$$

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

$$(x + \dfrac{1}{2} + \sqrt{(\dfrac{3i}{2}})) (x + \dfrac{1}{2} – \sqrt{(\dfrac{3i}{2}})) = 0$$

The roots of the given equation are $$\dfrac{-1}{2}$$ + $$\sqrt{(\dfrac{3i}{2})}$$, $$\dfrac{-1}{2}$$ – $$\sqrt{(\dfrac{3i}{2})^2}$$

Answered by Sakshi | 1 year ago

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