What will λ be for real and equal roots of (λ+1)x² − 2(λ−2)x + (λ+1) = 0?
Answer and explanation
Correct answer: λ = 1/2
For real and equal roots, the discriminant must be zero, and the coefficient of x² must remain non-zero. Here a = λ+1, b = −2(λ−2) and c = λ+1. Thus D = b² − 4ac = 4(λ−2)² − 4(λ+1)². Using the difference of squares, D = 4[(λ−2)² − (λ+1)²] = 4[(−3)(2λ−1)] = 12(1−2λ). Setting D = 0 gives 1−2λ = 0, so λ = 1/2. At this value, a = 3/2, which is non-zero, so the equation is genuinely quadratic and has equal real roots. Therefore option A is correct. λ = −1 would remove the quadratic term, while the other values do not make the discriminant zero.
Frequently asked questions
What is the correct answer to this question?
λ = 1/2
Why is this the correct answer?
For real and equal roots, the discriminant must be zero, and the coefficient of x² must remain non-zero. Here a = λ+1, b = −2(λ−2) and c = λ+1. Thus D = b² − 4ac = 4(λ−2)² − 4(λ+1)². Using the difference of squares, D = 4[(λ−2)² − (λ+1)²] = 4[(−3)(2λ−1)] = 12(1−2λ). Setting D = 0 gives 1−2λ = 0, so λ = 1/2. At this value, a = 3/2, which is non-zero, so the equation is genuinely quadratic and has equal real roots. Therefore option A is correct. λ = −1 would remove the quadratic term, while the other values do not make the discriminant zero.
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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