What's the conjugate root?

Algebra Level 4

One root of the equation x 2 + a x + b = 0 x^2+ax+b=0 is 3 + 8 \sqrt{3+\sqrt{8}}\; and a , b a,b are rational numbers .

Find the value of 15 a b 15ab .


The answer is 30.

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2 solutions

3 + 8 = 1 + 2 \sqrt{3+\sqrt8}=1+\sqrt2 .

As a a and b b are rational and one root is irrational, the other root must also be irrational and it must also be the conjugate of that root.

Thus the second root is 1 2 1-\sqrt2 .

Now we get the required quadratic by multiplying ( x ( 1 + 2 ) ) ( x ( 1 2 ) ) (x-(1+\sqrt2))(x-(1-\sqrt2)) , but an easier method is by using Vieta's Formula.

By Vieta's Formula, sum of the roots is a 1 = 1 + 2 + 1 2 \dfrac {-a} 1=1+\sqrt2+1-\sqrt2 or a = 2 a=-2 .

and product of the roots is b 1 = ( 1 + 2 ) ( 1 2 ) = 1 \dfrac {b} 1=(1+\sqrt2)(1-\sqrt2)=-1 or b = 1 b=-1 .

Thus 15 a b = 15 × 2 × 1 = 30 15ab=15 \times -2 \times -1=\boxed{30} .

I think you should add up why the other root must be conjugate of the first one.

Chaitnya Shrivastava - 5 years, 2 months ago

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because if it is not, then the product will of the roots will be irrational and so will the coefficients of the quadratic.

A Former Brilliant Member - 5 years, 2 months ago

I did the same.....

Aditya Kumar - 5 years, 1 month ago
Raz Lerman
Apr 16, 2016

Actually, by finding that (3 + 8^0.5)^0.5 = 1+ 2^0.5, allows you to subtitute that as the root of the quadratic equation, you know that the coefficient of x squared is 1, from there you subtitute 1 in A (in the original quadratic equation), from there, you know that -B(in the original quadratic equation) divided by 2 has to be 1(to be equal to 1 from 1 + 2^0.5)so you solve the linear equation -B÷2 = 1, where you find that B is equal to -2. you subtitute that in (B^2 - 4AC)^0.5 (from the original quadratic equation) where you are left with ((-2)^2 - 4 1 C)÷2 = 2^0.5, after you solved the linear equation you are left with C = -1, where in your question A is the coefficient of x^2, B is a and C is b. After that you do (-1) (-2) 15 = 30. sorry for my english, and if somebody doesn't get the solution, write it in the comments. *EDIT Some of the letters in my comment are messed up(- before the B in -B÷2 = 1. and it is 4 times 1 times C instead of 41C.)

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