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Coefficients add to zero, so 1 is a root and the other root is c/a

This type of question, asked 3 times: CDS 1 2023, CDS 2 2023, CDS 2 2025. All 3 questions below, oldest first.

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  1. Question 1

    CDS 1 2023

    12
    If a, b, c are non-zero real numbers such that a+b+c=0a + b + c = 0, then what are the roots of the equation ax2+bx+c=0ax^2 + bx + c = 0?

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  2. Question 2 (case study, 2 parts)

    CDS 2 2023

    A quadratic equation is given by (a+b+c)x2−(2a+2b)x+(a+b−c)=0(a + b + c)x^2 - (2a + 2b)x + (a + b - c) = 0; where a, b and c are real and distinct.
    49
    What are the roots of the equation ?

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    50
    Consider the following statements : 1. One of the roots of the equation is always less than 1 if a, b and c are all positive. 2. One of the roots of the equation is always negative if a, b and c are all negative. Which of the statements given above is/are correct ?

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  3. Question 3

    CDS 2 2025

    25
    If (a−b2)x2−(a+b2)x+b=0\left(\frac{a-b}{2}\right)x^2-\left(\frac{a+b}{2}\right)x+b=0, then what are the roots of this equation?

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