What condition on \(k\) is necessary for the equation \(x^2+2kx+16=0\) to have real roots?
Answer and explanation
Correct answer: \(k\leq -4\) or \(k\geq 4\)
For a quadratic equation \(ax^2+bx+c=0\) to have real roots, its discriminant must satisfy \(D=b^2-4ac\geq0\). Here, \(a=1, b=2k, c=16\), so \(D=(2k)^2-4(1)(16)=4k^2-64\). Thus, \(4k^2-64\geq0\Rightarrow k^2\\geq16\), which gives \(k\leq-4\) or \(k\geq4\). In option B, \(k\) lies between \(-4\) and \(4\), making the discriminant negative. Exam tip: Whenever real roots are asked for, begin with the condition \(D\geq0\).
Frequently asked questions
What is the correct answer to this question?
\(k\leq -4\) or \(k\geq 4\)
Why is this the correct answer?
For a quadratic equation \(ax^2+bx+c=0\) to have real roots, its discriminant must satisfy \(D=b^2-4ac\geq0\). Here, \(a=1, b=2k, c=16\), so \(D=(2k)^2-4(1)(16)=4k^2-64\). Thus, \(4k^2-64\geq0\Rightarrow k^2\\geq16\), which gives \(k\leq-4\) or \(k\geq4\). In option B, \(k\) lies between \(-4\) and \(4\), making the discriminant negative. Exam tip: Whenever real roots are asked for, begin with the condition \(D\geq0\).
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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