Find a quadratic polynomial, the sum and product of whose zeroes are -

1.	Find a quadratic polynomial, the sum and product of whose zeroes are -3/2√5 and -1/2, respectively. Also find its zeroes.
Answer» Sum of the zeroes = -3/2√5x Product of the zeroes = – ½ P(x) = x² – (sum of the zeroes) + (product of the zeroes) Then, P(x)= x² -3/2√5x – ½ P(x)= 2√5x² – 3x – √5 Using splitting the middle term method, 2√5x² – 3x – √5 = 0 2√5x² – (5x – 2x) – √5 = 0 2√5x² – 5x + 2x – √5 = 0 √5x (2x – √5) – (2x – √5) = 0 (2x – √5)(√5 – 1) = 0 ⇒ x = – 1/√5, √5/2

Find a quadratic polynomial, the sum and product of whose zeroes are -3/2√5 and -1/2, respectively. Also find its zeroes.

Answer»

Sum of the zeroes = -3/2√5x

Product of the zeroes = – ½

P(x) = x² – (sum of the zeroes) + (product of the zeroes)

Then, P(x)= x² -3/2√5x – ½

P(x)= 2√5x² – 3x – √5

Using splitting the middle term method,

2√5x² – 3x – √5 = 0

2√5x² – (5x – 2x) – √5 = 0

2√5x² – 5x + 2x – √5 = 0

√5x (2x – √5) – (2x – √5) = 0

(2x – √5)(√5 – 1) = 0

⇒ x = – 1/√5, √5/2