If the roots of a monic quadratic equation are r and 2r, and their sum is 9, what is the constant term?
Answer and explanation
Correct answer: 18
For a monic quadratic with roots α and β, the equation has the form x² − (α + β)x + αβ = 0. Consequently, the constant term is the product of the roots, not their sum. Here the roots are r and 2r, and their sum is 9, so r + 2r = 3r = 9. Hence r = 3, and the roots are 3 and 6. Their product is 3 × 6 = 18, which is therefore the constant term. Option B is the given sum and not the required product. Options C and D do not equal the product of the two roots. The monic condition is important because it ensures that the constant term itself, rather than a scaled version of it, equals the product of the roots. Thus option A is correct.
Frequently asked questions
What is the correct answer to this question?
18
Why is this the correct answer?
For a monic quadratic with roots α and β, the equation has the form x² − (α + β)x + αβ = 0. Consequently, the constant term is the product of the roots, not their sum. Here the roots are r and 2r, and their sum is 9, so r + 2r = 3r = 9. Hence r = 3, and the roots are 3 and 6. Their product is 3 × 6 = 18, which is therefore the constant term. Option B is the given sum and not the required product. Options C and D do not equal the product of the two roots. The monic condition is important because it ensures that the constant term itself, rather than a scaled version of it, equals the product of the roots. Thus option A is correct.
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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