Question

f (x)=ax^2+bx+c find alpa square/beta square +beta square/alpha square ​

Answer 1

$\huge{\boxed{\textbf{Answer :- }}}$

♦ Given that

$f(x) = ax^2 + bx + c$

♦ As we know that $\alpha \: and \beta$ are the roots of a quadratic equations

♦ Also we know that

>> Sum of roots = $\dfrac{-b}{a} = \alpha + \beta$

>> Product of roots = $\dfrac{c}{a} = \alpha \beta$

♦ Now to find

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

• We will first simplify and then substitute the value by a , b and c

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

$= \dfrac{\alpha^4 + \beta^4}{\alpha^2\beta^2}$

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

♦ By using values found above by Sum of roots and Product of roots.

$= \dfrac{[(\dfrac{-b}{a})^2 - 2\dfrac{c}{a}]^2 - 2(\dfrac{c}{a})^2}{(\dfrac{c}{a})^2}$

$= \dfrac{[\dfrac{b^2}{a^2} - \dfrac{2c}{a}]^2 - 2\dfrac{c^2}{a^2}}{\dfrac{c^2}{a^2}}$

$= \dfrac{[\dfrac{b^2 - 2ca}{a^2} ]^2 - 2\dfrac{c^2}{a^2}}{\dfrac{c^2}{a^2}}$

$= \dfrac{\dfrac{b^4 + 4c^2a^2 - 4b^2ca}{a^4} - 2\dfrac{c^2}{a^2}}{\dfrac{c^2}{a^2}}$

$=\dfrac{\dfrac{b^4 + 4c^2a^2 - 4b^2ca -2c^2a^2}{a^4}}{\dfrac{c^2}{a^2}}$

$= \dfrac{\dfrac{b^4 + 2c^2a^2 - 4b^2ca}{a^4}}{\dfrac{c^2}{a^2}}$

$= \dfrac{b^4 + 2c^2a^2 - 4b^2ca}{a^4} \times \dfrac{a^2}{c^2}$

$= \dfrac{b^4 + 2c^2a^2 - 4b^2ca}{a^2c^2}$