Find the zeroes of the quadratic polynomial 2x² - (1 + 2√(2))x + √(2) and verify the relationship between the zeroes and coefficents of the polynomial.

MathsCBSE Class 10 · Standard · Sample paper, session 2025-26 · 3 marks · Short answer

CBSE Class 10 Mathematics Standard sample paper 2025-26: question 28

From the official CBSE sample question paper for 2025-26, section C, question 28, for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q28 · 3 marks · Short answerPolynomials

Find the zeroes of the quadratic polynomial $2 x^{2} - (1 + 22) x + 2$ and verify the relationship between the zeroes and coefficents of the polynomial.

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2025-26.

Hints

Open one hint at a time and try again before the next.

Hint 1

Expand the middle term as $- x - 22 x$ .

Hint 2

Group the terms to find the common factor $(2 x - 1)$ .

Hint 3

Then compare the sum and the product of the zeroes with $- b / a$ and $c / a$ .

Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

Split the middle term: $2 x^{2} - (1 + 22) x + 2 = 2 x^{2} - x - 22 x + 2$ .
1 mark
$= x (2 x - 1) - 2 (2 x - 1) = (2 x - 1) (x - 2)$ .
So the zeroes are $\frac{1}{2}$ and $2$ .
1 mark
Verification ( $a = 2$ , $b = - (1 + 22)$ , $c = 2$ ):
Sum of zeroes $= \frac{1}{2} + 2 = \frac{1 + 2 2}{2} = \frac{- b}{a}$ .
Product of zeroes $= \frac{1}{2} \times 2 = \frac{2}{2} = \frac{c}{a}$ .
Hence the relationship is verified.
1 mark

Answer: Zeroes are 1/2 and √2; sum = −b/a and product = c/a verified

Where the marks are

Splits the middle term correctly (1 mark)
Factorises and gets the zeroes 1/2 and √2 (1 mark)
Verifies sum = −b/a and product = c/a (1 mark)

Revise the chapter

Chapter: Polynomials (notes, formulas, where marks are lost and every past question)

Practise this chapterBoard-style questions from Polynomials, one at a time, marked against the answer key and marking points.Back to the full paperSample paper 2025-26 (Standard): every question, by section.

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CBSE Class 10 Mathematics Standard sample paper 2025-26: question 28

Hints

Step-by-step solution

Revise the chapter

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