Question

if x=2+ root 3, then find the value of x^3-7x^2+13x+17

Answer 1

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♣♣ x³-7x²+13x+17

= (2+√3)³-7(2+√3)²+13(2+√3)+17

then solve it you will get the answer.

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Answer 2

Given:

x = 2 + √3.

To find:

The value of x^3 - 7x^2 + 13x + 17.

Solution:

As given, we have,

x = 2 + √3

So,

${x}^{2} = {(2 + √3)}^{2}$

$[as {(a + b)}^{2} = {a}^{2} + {b}^{2} +2ab]$

$= 4 + 3 + 4√3$

$= 7 + 4√3$

We need to find the value of the given expression

${x}^{3} - {7x}^{2} + 13x + 17$

This can be written as

${x}^{3} + 13x - {7x}^{2} + 17$

$= x( {x}^{2} + 13) - 7 {x}^{2} + 17$

Now,

after putting x = 2 + √3 and x^2 = 7 + 4 √3 in the above expression, we have,

$= 2 + √3(7 + 4 √3 + 13) - 7(7 + 4 √3) + 17$

$= 2 + √3(20 + 4 √3 ) - 49 - 28 √3) + 17$

$= 40 + 20√3 + 8√3 + 12 - 32 - 28 √3$

$= 20 + 28√3 - 28 √3$

$= 20$

Hence, the value of x^3 - 7x^2 + 13x + 17 is 20.