For the equation \(x^2-2(a-3)x+a^2-9=0\) to have real and equal roots, what should be the value of \(a\)?
Answer and explanation
Correct answer: \(a=3\)
A quadratic equation has real and equal roots only when its discriminant is zero. Here, \(D=[-2(a-3)]^2-4(a^2-9)=24(3-a)\). Therefore, \(24(3-a)=0\), giving \(a=3\). As a check, substituting \(a=3\) reduces the equation to \(x^2=0\), whose roots are equal. Exam tip: For equal roots, set the discriminant \(D\) equal to zero.
Frequently asked questions
What is the correct answer to this question?
\(a=3\)
Why is this the correct answer?
A quadratic equation has real and equal roots only when its discriminant is zero. Here, \(D=[-2(a-3)]^2-4(a^2-9)=24(3-a)\). Therefore, \(24(3-a)=0\), giving \(a=3\). As a check, substituting \(a=3\) reduces the equation to \(x^2=0\), whose roots are equal. Exam tip: For equal roots, set the discriminant \(D\) equal to zero.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Quadratic Equations. Topic: Nature of Roots.
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