If the roots of \(x^2-20x+k=0\) are real and distinct, what is the correct condition on \(k\)?
Answer and explanation
Correct answer: \(k<100\)
For a quadratic to have real and distinct roots the discriminant \(D=b^2-4ac\) must be positive. Here \(a=1,\;b=-20,\;c=k\), so
\(D=(-20)^2-4\cdot1\cdot k=400-4k\).
Real and distinct roots require \(D>0\), hence \(400-4k>0\) which gives \(k<100\).
If \(k=100\) the roots are equal, and if \(k>100\) the roots are complex. Exam tip: factor common multiples (\(D=4(100-k)\)) to check the sign quickly.
Frequently asked questions
What is the correct answer to this question?
\(k<100\)
Why is this the correct answer?
For a quadratic to have real and distinct roots the discriminant \(D=b^2-4ac\) must be positive. Here \(a=1,\;b=-20,\;c=k\), so
\(D=(-20)^2-4\cdot1\cdot k=400-4k\).
Real and distinct roots require \(D>0\), hence \(400-4k>0\) which gives \(k<100\).
If \(k=100\) the roots are equal, and if \(k>100\) the roots are complex. Exam tip: factor common multiples (\(D=4(100-k)\)) to check the sign quickly.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Quadratic Equations. Topic: Introduction to Quadratic Equations.
Student feedback
Was this question useful?
👍 0 Helpful 👎 0 Not helpful
Yes 0% No 0%
0 responsesStudent Reviews
No published reviews yet.
