If x² − (a + b + 1)x + ab + a + b = 0, on what does the nature of the roots depend for a and b?
Answer and explanation
Correct answer: D = (a − b)² + 1 − 2(a + b)
The governing concept is that the nature of the roots is determined by the discriminant D = B² − 4AC for a quadratic Ax² + Bx + C = 0. Comparing terms gives A = 1, B = −(a + b + 1), and C = ab + a + b. Thus D = (a + b + 1)² − 4(ab + a + b). Expanding and simplifying, D = a² + 2ab + b² + 2a + 2b + 1 − 4ab − 4a − 4b = a² − 2ab + b² + 1 − 2a − 2b = (a − b)² + 1 − 2(a + b). Therefore option A gives the correct expression. The other choices either omit terms or incorrectly claim that the discriminant is constant. Once this D is evaluated, D > 0, D = 0, or D < 0 identifies distinct real, equal, or non-real roots respectively.
Frequently asked questions
What is the correct answer to this question?
D = (a − b)² + 1 − 2(a + b)
Why is this the correct answer?
The governing concept is that the nature of the roots is determined by the discriminant D = B² − 4AC for a quadratic Ax² + Bx + C = 0. Comparing terms gives A = 1, B = −(a + b + 1), and C = ab + a + b. Thus D = (a + b + 1)² − 4(ab + a + b). Expanding and simplifying, D = a² + 2ab + b² + 2a + 2b + 1 − 4ab − 4a − 4b = a² − 2ab + b² + 1 − 2a − 2b = (a − b)² + 1 − 2(a + b). Therefore option A gives the correct expression. The other choices either omit terms or incorrectly claim that the discriminant is constant. Once this D is evaluated, D > 0, D = 0, or D < 0 identifies distinct real, equal, or non-real roots respectively.
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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