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Roots of a Quadratic Equation

#8203;Suppose...

Question

Suppose the quadratic polynomial P(x)=ax^2+bx+c has +ve coefficients a,b,c in AP in that order. If P(x)=0 has integer roots α and β then, α+β+αβ is?

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Solution

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$According to given condition a, b, c are in AP So, b - a = c - b . . . . (1) Since α and β are the roots of {ax}^{2} + bx + c = 0 So, α + β = - \frac{Coeff . of x}{Coeff . of x^{2}} = - \frac{b}{a} αβ = \frac{Constant term}{Coeff . of x^{2}} = \frac{c}{a} Now, α + β + αβ = - \frac{b}{a} + \frac{c}{a} = \frac{c - b}{a} = \frac{b - a}{a} [using 1] = \frac{b}{a} - 1$
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