Question

solve x^4-8x^3+18x^2-8x-7

Solve properly , I need an ans. worth 50 pts. ​

Answer 1

Answer:

x = 1 - √2

x = 3 - √2

x = 1 + √2

x = 3 + √2

Factorized answer = ( x² - 6 x + 7 )( x² - 2 x - 1 )

Step-by-step explanation:

For all the roots ( school method best )

x⁴ - 8 x³ + 18 x² - 8 x - 7 = 0

⇒ x⁴ - 2 x³ - x² - 6 x³ + 12 x² + 6 x + 7 x² - 14 x - 7 = 0

⇒ x² ( x² - 2 x - 1 ) - 6 x ( x² - 2 x - 1 ) + 7( x² - 2 x - 1 ) = 0

⇒ ( x² - 6 x + 7 )( x² - 2 x - 1 ) = 0

The factorized answer is required but I am also finding the roots of the given equation !

Apply Sridharacharya formula .

$x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$

You will find :

$x=\dfrac{6\pm\sqrt{36-28}}{2}\\\\\implies x=\dfrac{6\pm 2\sqrt{2}}{2}\implies x=3\pm\sqrt{2}\\\\x=\dfrac{2\pm \sqrt{4+4}}{2}\\\\\implies x=\dfrac{2\pm 2\sqrt{2}}{2}\implies x=1\pm \sqrt{2}$

x = 1 - √2

x = 3 - √2

x = 1 + √2

x = 3 + √2

Ignore this method . I was just experimenting with this ↓↓

Trial and error method

( quick for competitive exams [ but disadvantage is that it is applicable for finding one root only ] )

This is required when we are required to find only one root . Factorization takes a lot of time in such a case where the roots are rational .

There is no rational roots of the equation .We have to find the roots using the Bisection method .

Let f ( x ) = x⁴ - 8 x³ + 18 x² - 8 x - 7

Find the value of x such that f ( x ) is very close to zero .

x^4 - 8 x^3 + 18 x^2 - 8 x - 7

When x = 1

( 1 )^4 - 8 . 1 ^3 + 18 . 1^2 - 8.1 - 7

= > 1 - 8 + 36 - 8 - 7

= > 37 - 23

= > 14

When x = 0

( 0 )^4 - 8 . 0 ^3 + 18  

0^2 - 8.0 - 7

= > - 7

When x = - 1

f ( x ) = 28

Hence x must be between - 1 and 0 .

When x = - 0.5

( -0.5 )^4 - 8 . ( -0.5 ) ^3 + 18

( -0.5 )^2 - 8 ( - 0.5 ) - 7

= > 2.5626 .

This means x > - 0.5

When x = - 0.414

f ( x ) = ( -0.414 )^4 - 8 . ( -0.414 ) ^3 + 18 . ( -0.414 )^2 - 8 ( - 0.414 ) - 7

= > -0.005

This very close to 0.Hence we consider this as an approximate solution .One solution is - 0.414. Very carefully we note that x = √2 = 1.414 and hence x - 1 = 0.414.

Now this can be written as - ( √2 - 1 ) ⇒ 1 - √2 .

The answer is hence 1 - √2 ( this is for any one root ).

Other roots may exist . But different approach is needed ..... sigh .

Answer 2

Step-by-step explanation:

plzz refer to the attached file..

keep loving...❤️❤️❤️❤️✌️