Multiple choice

The nature of roots of the equation $\left( a+b+c \right) { x }^{ 2 }-2\left( a+b \right) x+\left( a+b-c \right) =0\left( a,b,c\epsilon Q \right) $

  1. Rational

  2. Irrational

  3. Imaginary

  4. None
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A Correct answer
Explanation

The discriminant D = [-2(a+b)]^2 - 4(a+b+c)(a+b-c) = 4(a+b)^2 - 4[(a+b)^2 - c^2] = 4(a+b)^2 - 4(a+b)^2 + 4c^2 = 4c^2. Since D = (2c)^2 is a perfect square, the roots are rational if the coefficients are rational.

AI explanation

To determine the nature of the roots of the quadratic equation (a + b + c)x^2 - 2(a + b)x + (a + b - c) = 0, we calculate its discriminant using the formula D = b^2 - 4ac. Substituting the coefficients gives D = [-2(a + b)]^2 - 4(a + b + c)(a + b - c). This simplifies to 4(a + b)^2 - 4[(a + b)^2 - c^2], which becomes 4c^2. Since 4c^2 is always non-negative and a perfect square, the roots of the equation are real and rational.

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