Multiple choice

Consider the quadratic equations $\displaystyle ax^{2}+2bx+c=0$ and $\displaystyle \left ( a+c \right )\left ( ax^{2}+2bx+c \right )-2\left ( ac-b^{2} \right )\left ( x^{2}+1 \right )=0$ If the roots of one are real (complex) then the roots of the other are complex (real.)

  1. True

  2. False
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A Correct answer
Explanation

The second equation is a transformation of the first. The relationship between the discriminants shows that if one has real roots, the other has complex roots, and vice versa.

AI explanation

The discriminant of the first equation is calculated as (2b) squared minus 4ac, which simplifies to 4b squared minus 4ac. Expanding and simplifying the second equation, (a plus c) times the first equation minus (2ac minus 2b squared) times (x squared plus 1), results in a new quadratic equation whose discriminant is calculated as 16(b squared minus ac) squared. Because the product of the discriminant of the first equation and the discriminant of the second equation is negative, if the roots of one equation are real, the roots of the other must be complex.

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