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

Asked by Sakshi | 1 year ago |  47

##### Solution :-

Given: x2 – x + 1 = 0

x2 – 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 + $$\dfrac{3}{4}$$ (-i)2 = 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

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

x = $$\dfrac{1}{2}$$ – $$\sqrt{\dfrac{3i}{2}}$$ or x = $$\dfrac{1}{2}$$ + $$\sqrt{\dfrac{3i}{2}}$$

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

Answered by Aaryan | 1 year ago

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