If the roots of the equation \(x^2+2(m-4)x+(m^2-7m+14)=0\) are real, which condition on \(m\) is correct?
Answer and explanation
Correct answer: \(m\leq 2\)
Here, \(a=1\), \(b=2(m-4)\), and \(c=m^2-7m+14\). Therefore, the discriminant is \(D=b^2-4ac=4(m-4)^2-4(m^2-7m+14)=4(2-m)\). For real roots, \(D\geq0\), so \(4(2-m)\geq0\), which gives \(m\leq2\). Remember that \(D=0\) gives two equal real roots, so the boundary value \(m=2\) is included.
Frequently asked questions
What is the correct answer to this question?
\(m\leq 2\)
Why is this the correct answer?
Here, \(a=1\), \(b=2(m-4)\), and \(c=m^2-7m+14\). Therefore, the discriminant is \(D=b^2-4ac=4(m-4)^2-4(m^2-7m+14)=4(2-m)\). For real roots, \(D\geq0\), so \(4(2-m)\geq0\), which gives \(m\leq2\). Remember that \(D=0\) gives two equal real roots, so the boundary value \(m=2\) is included.
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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