Question

prove that both roots of the equation x^2-x-3=0 are irrational

Answer 1

We have to prove that both the roots of the equation x² - x - 3 = 0 are irrational.

Proof : equation, x² - x - 3 = 0

⇒x² - 2(1/2)x + (1/2)² - (1/2)² - 3 = 0

⇒(x - 1/2)² - 1/4 - 3 = 0

⇒(x - 1/2)² - 13/4 = 0

⇒(x - 1/2)² = 13/4

Taking square root both sides we get

⇒x - 1/2 = ±√13/2

⇒x = (1 + √13)/2 and (1 - √13)/2

we if numbers are in the form of a ± √b, where a and b is rational number then numbers are irrational.

So here, (1 + √13)/2 = irrational

(1 - √13)/2 = irrational

Hence it is clear that both roots of equation x² - x - 3 are irrational.