If \(a\) is a real number and \(x^2-2(a+4)x+(a^2+8a+16)=0\), what is the nature of its roots?
Answer and explanation
Correct answer: Always real and equal
Here, \(A=1\), \(B=-2(a+4)\), and \(C=a^2+8a+16=(a+4)^2\). Therefore, the discriminant is \(D=B^2-4AC=4(a+4)^2-4(a+4)^2=0\). Hence, for every real value of \(a\), the roots are real and equal; in fact, the equation is \((x-(a+4))^2=0\), so the repeated root is \(x=a+4\). Exam tip: \(D=0\) indicates equal real roots.
Frequently asked questions
What is the correct answer to this question?
Always real and equal
Why is this the correct answer?
Here, \(A=1\), \(B=-2(a+4)\), and \(C=a^2+8a+16=(a+4)^2\). Therefore, the discriminant is \(D=B^2-4AC=4(a+4)^2-4(a+4)^2=0\). Hence, for every real value of \(a\), the roots are real and equal; in fact, the equation is \((x-(a+4))^2=0\), so the repeated root is \(x=a+4\). Exam tip: \(D=0\) indicates equal 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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