Question

Question is attached with image please solve!

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Answer 1

Answer:

Value of  x³ - 2x² - 7x + 5 = 16√8 + 49

Step-by-step explanation:

Given:

$\sf x=\dfrac{1}{3-\sqrt{8} }$

To Find:

The value of x³ - 2x² - 7x + 5

Solution:

By given,

$\sf x=\dfrac{1}{3-\sqrt{8} }$

Multiply the numerator and denominator with the conjugate,

$\sf \implies \dfrac{1}{3-\sqrt{8} } \times \dfrac{3+\sqrt{8} }{3+\sqrt{8} }$

Applying the identity (a + b) (a - b) = a² - b₂,

$\sf \implies \dfrac{3+\sqrt{8} }{3^2-(\sqrt{8} )^2}$

$\sf \implies \dfrac{3+\sqrt{8} }{9-8} =3+\sqrt{8}$

Now finding the value of x³ - 2x² - 7x + 5,

$\sf \implies (3+\sqrt{8} )^3-2\times(3+\sqrt{8} )^2-7\times(3+\sqrt{8})+5$

We know that

(a + b)³ = a³ + 3a²b + 3ab² + b³

(a + b)² = a² + 2ab + b²

Applying the identity,

$\implies \sf 3^3+3\times 3^2\times \sqrt{8} +3\times 3\times(\sqrt{8} )^2+(\sqrt{8})^3 -2(3^2+2\times 3\times \sqrt{8}+(\sqrt{8} )^2)-21-7\sqrt{8} +5$

$\sf \implies 27+27\sqrt{8} +72+8\times \sqrt{8} -2(9+6\sqrt{8} +8)-7\sqrt{8} -16$

$\sf \implies 83+27\sqrt{8} -18-12\sqrt{8} -16-7\sqrt{8}+8\sqrt{8}$

$\sf \implies 8\sqrt{8}+8\sqrt{8} +49$

$\sf \implies 16\sqrt{8} +49$

Hence the value of  x³ - 2x² - 7x + 5 is 16√8 + 49

Answer 2

Solution :

We have

$x = \frac{1}{3 - \sqrt{8} } = \frac{1}{3 - \sqrt{8} } \times \frac{(3 + \sqrt{8} )}{(3 + \sqrt{8}) } \\ \\ = \frac{(3 + \sqrt{8}) }{( {3})^{2} - ( \sqrt{8}) ^{2} } \\ \\ = \frac{(3 + \sqrt{8} )}{9 - 8} \\ \\ = (3 + \sqrt{8} ).$

x = 3 + √8 ⇒ x - 3 = √8

⇒ (x-3)² = (√8)² = 8

⇒ x² + 9 - 6x = 8

⇒ x² - 6x + 1 = 0

∴ x³ - 2x² - 7x + 5

= x (x²- 6x + 1) + 4 (x²-6x+1) +16x+1

= x × 0 + 4 × 0 +16 (3+√8)+1

= 48 + 16√8 + 1 = 49 + 32√2.