Question

If p, q are rational and (2^1÷2)-1 is a root of x2 – px + q = 0; pq =

Answer 1

question : If p and q are rational and (√2 - 1) is a root of x² - px + q = 0

To find : The value of pq

solution : it is given that (√2 - 1) is a root of polynomial x² - px + q = 0.

we know, if a root of quadratic equation is an irrational number then the other root of the quadratic equation is conjugate of given irrational number.

I mean, if (√a - b) is a root then (√a + b) will be another root.

so, (√2 + 1) is another root of x² - px + q = 0.

now sum of roots = -(-p)/1

⇒(√2 - 1) + (√2 + 1) = p

⇒2√2 = p

product of roots = q/1

⇒(√2 - 1)(√2 + 1) = q

⇒q = 2 - 1 = 1

so the value of pq = (2√2) × 1 = 2√2

Therefore the value of pq = 2√2

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