NSEJS #6

Algebra Level 3

If x = 1 2 − 3 x=\dfrac {1}{2-\sqrt{3}} , find the value of x 3 − 2 x 2 − 7 x + 5 x^3-2x^2-7x+5

This is the part of This set


The answer is 3.

Parth Lohomi by Parth Lohomi , 20, India

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3 solutions

Chew-Seong Cheong
Jun 22, 2015

x = 1 2 − 3 = 2 + 3 ( 2 − 3 ) ( 2 + 3 ) = 2 + 3 x=\dfrac{1}{2-\sqrt{3}} = \dfrac {2+\sqrt{3}}{(2-\sqrt{3})(2+\sqrt{3})} = 2 + \sqrt{3} .

This implies that x x is the root of x 2 − 4 x + 1 = 0 \space x^2 - 4x + 1 = 0 .

Therefore,

x 3 − 2 x 2 − 7 x + 5 = x 3 − 4 x 2 + x + 2 x 2 − 8 x + 2 + 3 = x ( x 2 − 4 x + 1 ) + 2 ( x 2 − 4 x + 1 ) + 3 = 0 + 0 + 3 = 3 \begin{aligned} x^3-2x^2-7x+5 & = x^3 - 4x^2 + x + 2x^2 - 8x + 2 + 3 \\ & = x(x^2 - 4x +1) +2(x^2-4x+1) + 3 \\ & = 0 +0 + 3 = \boxed{3} \end{aligned}

10 Helpful 1 Interesting 1 Brilliant 0 Confused

Sir, I solved it complete without factoring, and I am getting wrong answer:

x = 2 + 3 x=2+\sqrt{3} x 3 = 8 + 3 3 + 3 ( 2 ) ( 3 ) ( 2 + 3 ) = 8 + 3 3 + 6 3 ( 2 + 3 ) = 8 + 3 3 + 18 + 12 3 = 26 + 15 3 x^3=8+3\sqrt{3}+3(2)(\sqrt{3})(2+\sqrt{3})=8+3\sqrt{3}+6\sqrt{3}(2+\sqrt{3})=8+3\sqrt{3}+18+12\sqrt{3}=26+15\sqrt{3} 2 x 2 = 2 ( 5 + 4 3 ) = 10 + 8 3 2x^2=2(5+4\sqrt{3})=10+8\sqrt{3} 7 x = 14 + 7 3 7x=14+7\sqrt{3} ∴ x 3 − 2 x 2 − 7 x + 5 = 26 + 15 3 − 10 − 8 3 − 14 − 7 3 + 5 = 26 − 10 − 14 + 5 = 31 − 24 = 7 \therefore x^3-2x^2-7x+5=26+15\sqrt{3}-10-8\sqrt{3}-14-7\sqrt{3}+5=26-10-14+5=31-24=7

Can you please tell me my mistake? I am unable to find it. Thanks!

Vinayak Srivastava - 10 months, 4 weeks ago

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Also, I am unable to solve this problem . Can you please post a solution?

Vinayak Srivastava - 10 months, 4 weeks ago

Just use a calculator to check the calculations will do. 2 x 2 ≠ 10 + 8 3 2x^2 \ne 10+8\sqrt 3 .

Chew-Seong Cheong - 10 months, 4 weeks ago

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Oh, thank you! I forgot 2 2 ≠ 2 2^2 \neq 2 :-)

Vinayak Srivastava - 10 months, 4 weeks ago
Tanishq Varshney
Jun 21, 2015

x = 2 + 3 x=2+\sqrt{3}

The expression can be simplified as

x ( ( x − 2 ) x − 7 ) + 5 x((x-2)x-7)+5

replacing value of x x

( 2 + 3 ) ( 3 ( 2 + 3 ) − 7 ) + 5 (2+\sqrt{3})(\sqrt{3}(2+\sqrt{3})-7)+5

− 2 ( 2 2 − ( 3 ) 2 ) + 5 -2(2^{2}-(\sqrt{3})^{2})+5

− 2 + 5 = 3 -2+5=3

2 Helpful 0 Interesting 0 Brilliant 0 Confused
Aakash Khandelwal
Jun 21, 2015

in your NSEJS#5 answer should be 25/3

as by squaring and adding we get a^2 +b^2=25

i am sorry for discussing that problem in this one as i am unable to ask there as no one has gave there solution until now. @Parth Lohomi

2 Helpful 0 Interesting 0 Brilliant 0 Confused

Yeah Correct bro even I am getting 25 but I am very frustrated that I can't type the answer which is 8.33 as it doesn't except non-integer values.

Kushagra Sahni - 5 years, 11 months ago

What do you mean?

a 2 + b 2 = 25 a^2 + b^2 = 25 is the answer.

Kishore S. Shenoy - 5 years, 9 months ago

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