Q)  Find a quadratic polynomial whose zeroes are (5-2√3) and (5+2√3).

(Q24 – 30/1/3 – CBSE 2026 Question Paper)

Ans: 

Step 1: The zeroes for the polynomial  given as (5 – 2√3) and (5 + 2√3):

Let’s consider, one root, α = (5 – 2√3) and other root, β = (5 + 2√3)

Step 2: Sum of the zeroes of polynomial:

α + β = (5 – 2√3) + (5 + 2√3) = 5 – 2√3 + 5 + 2√3

∴ α + β = 10 …………. (i)

Step 3: Product of the zeroes of polynomial:

α . β = (5 – 2√3) x (5 + 2√3) ……… (ii)

∵ by algebraic identity: (a – b) (a + b) = a² – b²

∴ (5 – 2√3) x (5 + 2√3) = (5)² – (2√3)²

= 25 – 12 = 13

Substituting this value in equation (ii), we get:

α . β  = 13 ………….. (iii)

Step 4: ∵ Quadratic polynomial is given by:

f (x) = k [x² – (sum of the zeroes) x + (product of the zeroes)]

Here, k is  non-zero constant (any real number).

∴ f (x) = k [x² – (α + β) x + (α . β)]

Substituting values from equation (i) and equation (iii), we get:

∴ f (x) = k [x² – (10) x + (13)]

∴ f (x) = k [x² – 10 x + 13]

For f (x) = 0, since k ≠ 0; 

∴  x² – 10 x + 13 = 0

Hence, the required quadratic polynomial is x² – 10 x + 13 = 0.

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