Question

$\underline{ \boxed{\mathbb{\red{question}}}} \\ \\ p \: and \: q are \: zeros \: of \: {3x}^{2} + 2 x - 9 \: then \: the \: value \: of \: \: P \: - Q \: \: is$ ${ \boxed{\mathbb{\pink{ reward \: for \: correct \: answer}}}}$​

Answer 1

$\large\underline{\sf{Solution-}}$

$\rm :\longmapsto\:p \: and \: q \: are \: zeroes \: of \: {3x}^{2} + 2x - 9$

We know that,

$\boxed{\red{\sf Sum\ of\ the\ zeroes=\frac{-coefficient\ of\ x}{coefficient\ of\ x^{2}}}}$

$\bf\implies \:p + q \: = \: - \: \dfrac{2}{3}$

And

$\boxed{\red{\sf Product\ of\ the\ zeroes=\frac{Constant}{coefficient\ of\ x^{2}}}}$

$\bf\implies \:p q \: = \: - \: \dfrac{9}{3} = \: - \: 3$

Now, Consider

$\rm :\longmapsto\:p - q$

can be rewritten as

$\rm \:  =  \: \sqrt{ {(p - q)}^{2} }$

$\rm \:  =  \: \sqrt{ {p}^{2} + {q}^{2} - 2pq}$

$\rm \:  =  \: \sqrt{ {p}^{2} + {q}^{2} + 2pq - 4pq}$

$\rm \:  =  \: \sqrt{ {(p + q)}^{2} - 4pq}$

$\rm \:  =  \: \sqrt{ {\bigg[ - \dfrac{2}{3} \bigg]}^{2} - 4 \times ( - 3)}$

$\rm \:  =  \: \sqrt{\dfrac{4}{9} + 12}$

$\rm \:  =  \: \sqrt{\dfrac{4 + 108}{9}}$

$\rm \:  =  \: \sqrt{\dfrac{112}{9}}$

$\rm \:  =  \: \sqrt{\dfrac{4 \times 4 \times 7}{3 \times 3} }$

$\rm \:  =  \: \dfrac{4}{3} \sqrt{7}$

$\rm\implies \:\boxed{\tt{ p - q = \frac{4}{3} \sqrt{7}}}$

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Additional Information

$\red{\rm :\longmapsto\: \alpha , \beta , \gamma \: are \: zeroes \: of \: a {x}^{3} + b {x}^{2} + cx + d, \: then}$

$\boxed{ \bf{ \: \alpha + \beta + \gamma = - \dfrac{b}{a}}}$

$\boxed{ \bf{ \: \alpha \beta + \beta \gamma + \gamma \alpha = \dfrac{c}{a}}}$

$\boxed{ \bf{ \: \alpha \beta \gamma = - \dfrac{d}{a}}}$

Answer 2

Answer:

Determine the concentration and total amount of glucose and (NH4)2SO4 in the nutrient medium. b). Determine the yield coefficients YXS and YXO2. c). Determine the total amount of oxygen required