Question

If alpha and bita are the roots of x^2=x-1 then the value of alpha^2/bita-bita^2/alpha is​

Answer 1

Answer:

• → 3

Step-by-step explanation:

The given equation is,

x² = x - 1

→ x² - x + 1 = 0, it is a quadratic equation

We know, the common structure of quadratic equation,

ax² + bx + c = 0

In the given question,

• a = 1
• b = ( -1 )
• c = 1

If α and β are the roots of the equation, then

→ Sum of the roots,

$\alpha + \beta = - \dfrac{ b}{a}$

$\implies \: - \dfrac{ (-1)}{1} = 1$

→ Product of roots,

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

$\implies \: \dfrac{1}{1} = 1$

Now,

$\sf \: \dfrac{ { \alpha }^{2} }{ \beta } - \dfrac{ { \beta }^{2} }{ \alpha }$

$\implies \: \dfrac{ { \alpha }^{3} - { \beta }^{3} }{ \alpha \beta }$

$\implies \: \dfrac{ - {[( \beta + \alpha )^{3} - 3 \alpha \beta ( \alpha + \beta)}] }{ \alpha \beta }$

$\implies \: \dfrac{ - {[( 1)^{3} - 3 \times 1 \times 1}] }{ 1 }$

$\implies \: 3$