What is the nature of the roots of x² − 7x + 11 = 0?
Answer and explanation
Correct answer: Two real, irrational, and distinct roots (D = 5)
Use the discriminant D = B² − 4AC to classify the roots. In x² − 7x + 11 = 0, A = 1, B = −7, and C = 11. Hence D = (−7)² − 4(1)(11) = 49 − 44 = 5. Since D > 0, there are two real and distinct roots. Since 5 is not a perfect square, √5 is irrational; the quadratic formula gives roots (7 + √5)/2 and (7 − √5)/2, both irrational. Therefore option A is correct. Option B mistakes the discriminant for B² and also incorrectly calls the roots rational. Equal roots require D = 0, while no real roots require D < 0, so options C and D do not apply.
Frequently asked questions
What is the correct answer to this question?
Two real, irrational, and distinct roots (D = 5)
Why is this the correct answer?
Use the discriminant D = B² − 4AC to classify the roots. In x² − 7x + 11 = 0, A = 1, B = −7, and C = 11. Hence D = (−7)² − 4(1)(11) = 49 − 44 = 5. Since D > 0, there are two real and distinct roots. Since 5 is not a perfect square, √5 is irrational; the quadratic formula gives roots (7 + √5)/2 and (7 − √5)/2, both irrational. Therefore option A is correct. Option B mistakes the discriminant for B² and also incorrectly calls the roots rational. Equal roots require D = 0, while no real roots require D < 0, so options C and D do not apply.
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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