Question

If alpha and beta are the zeros of the polynomial 2x2+7x+5,then alpha square by beta +beta square by alpha is

Answer 1

Answer:

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Answer 2

QUESTION

If α and β are the zeroes of the polynomial 2x² + 7x + 5 ,  then find

$\frac{\alpha ^{2} }{\beta } + \frac{\beta ^{2} }{\alpha }$ .

SOLUTION

Comparing the given quadratic polynomial with the standard form of quadratic polynomial i.e,, ax² + bx + c

we will get,

a = 2 ; b = 7 ; c = 5

SO,

α+β = -b /a = - 7 / 2   _____equation (1)

αβ = c / a = 5 / 2   _______equation (2)

We have to find

$\frac{\alpha ^{2} }{\beta } + \frac{\beta ^{2} }{\alpha } \\\\( taking LCM )\\\\\frac{\alpha ^{3}+\beta ^{3} }{\alpha\beta }$

[ By identity

( x+y)³ = x³ + y³ + 3 xy ( x + y)

we know,

x³ + y³ = (x + y)³ - 3 xy ( x + y ) ]

so,

$\frac{\alpha ^{3}+\beta^{3} }{\alpha\beta } \\\\=\frac{(\alpha+\beta)^{3}-3\alpha\beta(\alpha + \beta) }{\alpha\beta }$

putting equation (1) and (2)

we will get,

$\frac{(\frac{-7}{2})^{3} - 3 (\frac{5}{2})(\frac{-7}{2} ) }{\frac{5}{2} } \\\\=\frac{\frac{-343}{8}-3(\frac{-35}{4} ) }{\frac{5}{2} } \\\\=\frac{\frac{-343}{8}+\frac{105}{4} }{\frac{5}{2} } \\\\=\frac{\frac{-343 + 210}{8} }{\frac{5}{2} } \\\\=\frac{-133}{8} * \frac{2}{5} \\\\=\frac{-133}{20}$

HENCE,

α² / β + β² /α  = -133/20.