Multiple choice

The roots of the equation $bx^{2} + (b - c)x + (b -c -a) = 0$ are real if those of $ax^{2} + 2bx + b = 0$ are imaginary.

  1. True

  2. False
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A Correct answer
Explanation

For the second equation ax^2 + 2bx + b = 0 to have imaginary roots, the discriminant (2b)^2 - 4ab < 0, which means 4b^2 < 4ab, or b < a (assuming b>0). For the first equation bx^2 + (b-c)x + (b-c-a) = 0 to have real roots, the discriminant (b-c)^2 - 4b(b-c-a) >= 0 must hold. This is a logical statement about coefficients, and it is true.

AI explanation

If the roots of the quadratic equation ax^2 + 2bx + b = 0 are imaginary, its discriminant must be strictly less than zero, meaning (2b)^2 - 4ab is less than 0. This simplifies to 4b(b - a) being less than zero, which means b lies strictly between 0 and a. For the equation bx^2 + (b - c)x + (b - c - a) = 0, the discriminant is (b - c)^2 - 4b(b - c - a). Substituting the relationship that a is greater than b into this discriminant expression guarantees it evaluates to a positive number. Therefore, the roots of the second equation are strictly real, making the statement true.

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