Question

If α and β are the zeroes of the polynomial x² – 5x + k such that α – β = 1, find the value of k.

Concepts Used

This question tests your understanding of:

  • Sum and product of zeroes of a quadratic polynomial
  • Solving a pair of linear equations in two variables
  • Relationship between coefficients and zeroes

Step-by-Step Solution

Step 1: Identify the coefficients

Given polynomial: x² – 5x + k. Comparing with standard form ax² + bx + c:

a = 1, b = –5, c = k

Step 2: Apply sum of zeroes formula

α + β = –b/a = –(–5)/1 = 5

Step 3: Apply product of zeroes formula

αβ = c/a = k/1 = k

Step 4: Use the given condition

α – β = 1

Step 5: Add the two equations

(α + β) + (α – β) = 5 + 1

2α = 6

α = 3

Step 6: Find β

α + β = 5

3 + β = 5

β = 2

Step 7: Find k

k = αβ = 3 × 2 = 6

✓ Answer

k = 6

Verification

Substitute x = 3 and x = 2 back into the polynomial:

For α = 3: (3)² – 5(3) + 6 = 9 – 15 + 6 = 0 ✓

For β = 2: (2)² – 5(2) + 6 = 4 – 10 + 6 = 0 ✓

Both values satisfy the polynomial. Hence k = 6 is correct. ✅

Video Solution

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Practice Questions

Try solving these similar questions:

  1. If α and β are zeroes of x² – 6x + k such that α – β = 2, find k. (Answer: k = 8)
  2. If α and β are zeroes of 2x² + 5x + k such that α + β + αβ = –1, find k. (Answer: k = –3)

Also Appears In

This question is commonly asked in:

  • CBSE Class 10 Board Exams (2-3 marks)
  • SSC CGL Tier 1 (Algebra section)
  • RRB NTPC (Mathematics section)