Question

if the sum of the roots of the quadratic equation X square + bx + c is equal to zero is equal to the sum of the squares of their reciprocals then prove that to a square C is equal to C square b + b square a
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Answer 1

$bc^{2}, a^{2} c$ and $b^{2}a$ are in A.P.

Step-by-step explanation:

The given quadratic equation:

$ax^{2} +bx+c=0$

Let α and  β are the roots of quadratic equation.

∴ α + β = $\dfrac{-b}{a}$ and

$\alpha \beta = \dfrac{c}{a}$

According to question,

$\alpha+ \beta =\dfrac{1}{\alpha^{2} } +\dfrac{1}{\beta^{2}}$

$=\dfrac{\alpha ^{2}+ \beta^{2}  }{\alpha^{2} \beta^{2}}$

$=\dfrac{(\alpha +\beta )^{2}-2\alpha \beta }{(\alpha \beta)^{2} }$

∴ $\dfrac{-b}{a} =\dfrac{(\dfrac{-b}{a})^{2} -2ac}{(\dfrac{c}{a} )^{2} }$

⇒ $-bc^{2} =b^{2} a-2a^{2} c$

⇒ $a^{2} c-bc^{2} =b^{2}a -a^{2} c$

∴ $bc^{2}, a^{2} c$ and $b^{2}a$ are in A.P.

Hence, it is proved.