If \(a\) is a real parameter and \(x^2-2(a+4)x+a^2+8a+15=0\), what will be the nature of its roots?
Answer and explanation
Correct answer: The roots will always be real and distinct; they will also be rational when \(a\) is rational
Here \(A=1\), \(B=-2(a+4)\), and \(C=a^2+8a+15\). Therefore, the discriminant is \(D=B^2-4AC=4(a+4)^2-4(a^2+8a+15)=4>0\). Hence, the roots are always real and distinct. In fact, the roots are \(a+3\) and \(a+5\); thus, they are rational when \(a\) is rational. Option B is incorrect because \(D\neq0\), and option D is incorrect because rational values of \(a\) give rational roots. Exam tip: \(D>0\) indicates real and distinct roots, \(D=0\) indicates equal roots, and \(D<0\) indicates non-real roots.
Frequently asked questions
What is the correct answer to this question?
The roots will always be real and distinct; they will also be rational when \(a\) is rational
Why is this the correct answer?
Here \(A=1\), \(B=-2(a+4)\), and \(C=a^2+8a+15\). Therefore, the discriminant is \(D=B^2-4AC=4(a+4)^2-4(a^2+8a+15)=4>0\). Hence, the roots are always real and distinct. In fact, the roots are \(a+3\) and \(a+5\); thus, they are rational when \(a\) is rational. Option B is incorrect because \(D\neq0\), and option D is incorrect because rational values of \(a\) give rational roots. Exam tip: \(D>0\) indicates real and distinct roots, \(D=0\) indicates equal roots, and \(D<0\) indicates non-real roots.
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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