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If α and β are zeros of the quadratic polynomial 2x2+3x−6, then find the values of α2+β2.

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\frac{30}{7}

\frac{31}{7}

\frac{33}{4}

\frac{33}{2}

answer is C.

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Detailed Solution

Given polynomial 2x2 + 3x − 6
Roots of the above polynomial (quadratic expression) is α and β.
So, as in quadratic polynomial:
Sum of roots =

\frac{- b}{a}

= α+β
Product of roots =

\frac{c}{a}

= αβ
Now, in case of our polynomial:
i.e. 2x2+3x−6 on comparing with ax2 + bx + c
Here, a = 2; b = 3; c = −6
So, Sum of roots =

\frac{- b}{a}

\frac{- 3}{2}

= α+β
Product of roots =

\frac{c}{a}

\frac{- 6}{2}

= −3 = αβ
Now we have to find out the value of α2 + β2
As we know, (a+b)2 = a2 + b2 + 2ab
⇒ a2 + b2 = (a+b)2 − 2ab
So, Putting the value of a = α, b = β,
We get,
α2 + β2 = (α+β)2 − 2αβ …..(i)
In our case, α+β =

\frac{- 3}{2}

αβ = −3
So, putting (α+β) and αβ in equation (i)
We get,
α2 + β2 =

{(\frac{- 3}{2})}^{2}

− 2(−3)
⇒ α2 + β2 =

\frac{9}{4}

+ 6
⇒ α2 + β2 =

\frac{9 + 24}{4}

⇒ α2 + β2 =

\frac{33}{4}

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