What is the discriminant of the equation \(x^2+7x+10=0\), and what is the nature of its roots?
Answer and explanation
Correct answer: D = 9; two real, rational and distinct roots
For a quadratic equation \(ax^2+bx+c=0\), the discriminant is \(D=b^2-4ac\). Here, \(a=1, b=7, c=10\), so \(D=7^2-4(1)(10)=49-40=9\). Since \(D\) is positive and a perfect square, the roots are real, rational, and distinct; in fact, they are \(-5\) and \(-2\). Therefore, option A is correct. Exam tip: \(D>0\) gives distinct real roots, and a perfect-square discriminant makes those roots rational.
Frequently asked questions
What is the correct answer to this question?
D = 9; two real, rational and distinct roots
Why is this the correct answer?
For a quadratic equation \(ax^2+bx+c=0\), the discriminant is \(D=b^2-4ac\). Here, \(a=1, b=7, c=10\), so \(D=7^2-4(1)(10)=49-40=9\). Since \(D\) is positive and a perfect square, the roots are real, rational, and distinct; in fact, they are \(-5\) and \(-2\). Therefore, option A is correct. Exam tip: \(D>0\) gives distinct real roots, and a perfect-square discriminant makes those roots rational.
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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