Question

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21. Ghen that a and B are the roots of the equation r = 7x +4,
a show that a³ = 53 +28
(b) find the value of a/ b+b/a
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pls this is urgent anybody help me​

Answer 1

As the roots are a and b, we only required values in a.

So, Substitute x = a (as it is a solution of it.)

Then,

x² = 7x + 4 ______1

Multiply "a" on both sides,

x³ = 7x² + 4x ______2

Substituting 1 on 2

We get,

x³ = 7(7x +4) + 4x

x³ = 49x + 28 +4x

x³ = 53x + 4x

(HENCE PROVED)

Answer 2

$\bold{x^2−7x+4 }$

$\bold{⟹x^2−7x−4=0}$

$\bold{α+β=7}$

$\bold{αβ=−4}$

$\bold{1.⟹α^2−7α−4=0}$

$\bold{⟹(α−7)(α^2 −7α−4)=0}$

$\small \bold{⟹α^3 −7α^2−4α +7α^2−49α−28=0}$

$\bold{⟹α^3−53α−28=0}$

$\bold{⟹α^3=53α+28}$

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

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

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

$= \frac{49 -2( - 4)}{ - 4}$

$= \frac{49 + 8}{ - 4}$

$= - \frac{57}{4}$