Question

If α and β are the zeroes of the quadratic polynomial 3x²+5x+2, find the value of α² and β² .

Answer 1

hope there is (+) instead of "and"

Answer 2

$\large\underline\mathrm\red{Given:-}$

• $\large\mathrm{p(x) = {3x}^{2} - 5x + 7}$

$\large\underline\mathrm\red{To \: find}$

• $\large\mathrm{Value \: of \: {α}^{2} \: + \: {β}^{2}}$

$\large\underline\mathrm\red{Solution}$

$\large\mathrm{p(x) = {3x}^{2} - {5x} +7}$

• $\large\mathrm{a = 3}$

• $\large\mathrm{b = -5}$

• $\large\mathrm{c = 7}$

$\large\underline\mathrm\red{Let \: {α} \: and \: {β} \: are \: the \: zeroes \: of \: the \: given \: polynomial.}$

$\large\mathrm{Sum \: of \: zeroes \: = \: -b/a}$

$\large\mathrm{⟹ {α} \: + \: {β} \: = \: -(-5)/3}$

$\large\mathrm{⟹ {α} + {β} = 5/3}$

$\large\mathrm{Product \: of \: zeroes = c/a}$

$\large\mathrm{⟹ {αβ} = 7/3}$

$\large\underline\mathrm\red{we \: know \: that,}$

$\large\mathrm{⟹ {(a+b)}^{2} = {a}^{2} + {b}^{2} + {2ab}}$

$\large\mathrm{⟹ {(a+b)}^{2} - {2ab} = {a}^{2} + {b}^{2}}$

$\large\mathrm{So, {α}^{2} + {β}^{2} = ({α} + {β})^{2} - {2αβ}}$ ..(i)

$\large\mathrm\red{Putting \: values \: of \: {α} + {β} = 5/3 \: and \: {αβ} = 7/3 \: in \: equation \: (i).</p><p>}$

$\large\mathrm{{α}^{2} + {β}^{2} = {(5/3)}^{2} - 2× 7/3}$

$\large\mathrm{{α}^{2} + {β}^{2} = 25/9 -14/3}$

$\large\mathrm{{α}^{2} + {β}^{2} = (25- 42)/9}$

$\large\mathrm{{α}^{2} + {β}^{2} = -17/9}$

$\large\mathrm\red{So, \: value \: of \: { \alpha }^{2} + { \beta }^{2} = {-17}/{9}}$

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