if alpha and beta are the zeroes of the quadratic polynomial x^2 - 6x + x then find the value of a, if 3alpha + 2beta = 24
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![p(x) = {x}^{2} - 6x + a \\ we \: know \: a {x}^{2} + bx + c \\ \\ a = 1 || b = - 6 || c = a \\ \\ \alpha + \beta(sum \: of \: zeroes) = \frac{ - b}{a} \\ \alpha + \beta = 6 \\ \\ \alpha \beta(product \: of \: zeroes) = \frac{c}{a} \\ \alpha \beta = a \\ \\ = > 3 \alpha + 2 \beta = 24 \\ = > 3 \alpha - \alpha + 2 \beta - \beta = 24 - 6 \\ = > 2 \alpha + \beta = 18 \\ \\ = > 2 \alpha - \alpha + \beta - \beta = 18 - 6 \\ \alpha = 12 \\ \\ \alpha + \beta = 6 \\ \beta = - 6 \\ \\ \alpha \beta = a \\ = > 12 \times ( - 6) = a \\ = > - 72 = a \\ \\ hence \: a = - 72 p(x) = {x}^{2} - 6x + a \\ we \: know \: a {x}^{2} + bx + c \\ \\ a = 1 || b = - 6 || c = a \\ \\ \alpha + \beta(sum \: of \: zeroes) = \frac{ - b}{a} \\ \alpha + \beta = 6 \\ \\ \alpha \beta(product \: of \: zeroes) = \frac{c}{a} \\ \alpha \beta = a \\ \\ = > 3 \alpha + 2 \beta = 24 \\ = > 3 \alpha - \alpha + 2 \beta - \beta = 24 - 6 \\ = > 2 \alpha + \beta = 18 \\ \\ = > 2 \alpha - \alpha + \beta - \beta = 18 - 6 \\ \alpha = 12 \\ \\ \alpha + \beta = 6 \\ \beta = - 6 \\ \\ \alpha \beta = a \\ = > 12 \times ( - 6) = a \\ = > - 72 = a \\ \\ hence \: a = - 72](https://tex.z-dn.net/?f=p%28x%29+%3D+%7Bx%7D%5E%7B2%7D+-+6x+%2B+a+%5C%5C+we+%5C%3A+know+%5C%3A+a+%7Bx%7D%5E%7B2%7D+%2B+bx+%2B+c+%5C%5C+%5C%5C+a+%3D+1+%7C%7C+b+%3D+-+6+%7C%7C+c+%3D+a+%5C%5C+%5C%5C+%5Calpha+%2B+%5Cbeta%28sum+%5C%3A+of+%5C%3A+zeroes%29+%3D+%5Cfrac%7B+-+b%7D%7Ba%7D+%5C%5C+%5Calpha+%2B+%5Cbeta+%3D+6+%5C%5C+%5C%5C+%5Calpha+%5Cbeta%28product+%5C%3A+of+%5C%3A+zeroes%29+%3D+%5Cfrac%7Bc%7D%7Ba%7D+%5C%5C+%5Calpha+%5Cbeta+%3D+a+%5C%5C+%5C%5C+%3D+%26gt%3B+3+%5Calpha+%2B+2+%5Cbeta+%3D+24+%5C%5C+%3D+%26gt%3B+3+%5Calpha+-+%5Calpha+%2B+2+%5Cbeta+-+%5Cbeta+%3D+24+-+6+%5C%5C+%3D+%26gt%3B+2+%5Calpha+%2B+%5Cbeta+%3D+18+%5C%5C+%5C%5C+%3D+%26gt%3B+2+%5Calpha+-+%5Calpha+%2B+%5Cbeta+-+%5Cbeta+%3D+18+-+6+%5C%5C+%5Calpha+%3D+12+%5C%5C+%5C%5C+%5Calpha+%2B+%5Cbeta+%3D+6+%5C%5C+%5Cbeta+%3D+-+6+%5C%5C+%5C%5C+%5Calpha+%5Cbeta+%3D+a+%5C%5C+%3D+%26gt%3B+12+%5Ctimes+%28+-+6%29+%3D+a+%5C%5C+%3D+%26gt%3B+-+72+%3D+a+%5C%5C+%5C%5C+hence+%5C%3A+a+%3D+-+72)
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