If the roots of the equation \(x^2-8x+k=0\) are real and distinct, what is the correct condition on \(k\)?
Answer and explanation
Correct answer: \(k<16\)
For a quadratic equation to have real and distinct roots, its discriminant \(D=b^2-4ac\) must be positive. Here, \(a=1, b=-8, c=k\), so \(D=(-8)^2-4(1)(k)=64-4k\). Thus, \(64-4k>0\), which gives \(k<16\). When \(k=16\), the roots are equal, so option B is incorrect. Exam tip: use \(D>0\) specifically for distinct real roots.
Frequently asked questions
What is the correct answer to this question?
\(k<16\)
Why is this the correct answer?
For a quadratic equation to have real and distinct roots, its discriminant \(D=b^2-4ac\) must be positive. Here, \(a=1, b=-8, c=k\), so \(D=(-8)^2-4(1)(k)=64-4k\). Thus, \(64-4k>0\), which gives \(k<16\). When \(k=16\), the roots are equal, so option B is incorrect. Exam tip: use \(D>0\) specifically for distinct real roots.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Quadratic Equations. Topic: Introduction to Quadratic Equations.
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