For which value of \(k\) will \(x^2-6x+k\) have real and irrational zeroes?
Answer and explanation
Correct answer: k = 7
Use the discriminant: \(D=b^2-4ac=36-4k=4(9-k)\). For real roots we need \(D\ge0\) (i.e. \(k\le9\)). For the roots to be irrational we need \(D>0\) and \(D\) not a perfect square.
- For \(k=7\), \(D=36-28=8\), which is positive and not a perfect square, so the roots are real and irrational (correct).
- For \(k=5\), \(D=16\) which is a perfect square, giving rational roots.
- For \(k=9\), \(D=0\) so the roots are equal and rational.
- For \(k=10\), \(D=-4\) is negative, so the roots are complex (not real).
Exam tip: Always check that \(D>0\) and then test whether \(D\) is a perfect square to decide irrational vs rational roots.
Frequently asked questions
What is the correct answer to this question?
k = 7
Why is this the correct answer?
Use the discriminant: \(D=b^2-4ac=36-4k=4(9-k)\). For real roots we need \(D\ge0\) (i.e. \(k\le9\)). For the roots to be irrational we need \(D>0\) and \(D\) not a perfect square.
- For \(k=7\), \(D=36-28=8\), which is positive and not a perfect square, so the roots are real and irrational (correct).
- For \(k=5\), \(D=16\) which is a perfect square, giving rational roots.
- For \(k=9\), \(D=0\) so the roots are equal and rational.
- For \(k=10\), \(D=-4\) is negative, so the roots are complex (not real).
Exam tip: Always check that \(D>0\) and then test whether \(D\) is a perfect square to decide irrational vs rational roots.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Polynomials. Topic: Irrational numbers and real numbers.
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