If the difference between the roots of the equation \(x^2-(k+2)x+2k=0\) is 2, find the values of \(k\).
Answer and explanation
Correct answer: 0 and 4
For roots \(\alpha\) and \(\beta\) of \(ax^2+bx+c=0\), \((\alpha-\beta)^2=\frac{b^2-4ac}{a^2}\). Here, \(a=1, b=-(k+2), c=2k\), so \((\alpha-\beta)^2=(k+2)^2-8k=(k-2)^2\). Since the difference is 2, \((k-2)^2=4\), giving \(k-2=\pm2\), hence \(k=0\) or \(k=4\). Therefore, option A is correct. Exam tip: Relate the square of the difference between the roots to the discriminant before solving for the parameter.
Frequently asked questions
What is the correct answer to this question?
0 and 4
Why is this the correct answer?
For roots \(\alpha\) and \(\beta\) of \(ax^2+bx+c=0\), \((\alpha-\beta)^2=\frac{b^2-4ac}{a^2}\). Here, \(a=1, b=-(k+2), c=2k\), so \((\alpha-\beta)^2=(k+2)^2-8k=(k-2)^2\). Since the difference is 2, \((k-2)^2=4\), giving \(k-2=\pm2\), hence \(k=0\) or \(k=4\). Therefore, option A is correct. Exam tip: Relate the square of the difference between the roots to the discriminant before solving for the parameter.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Quadratic Equations. Topic: Roots of a Quadratic Equation.
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