An algebra problem by Anik Mandal

Algebra Level 3

$\large a+a^2+a^3 = 1 = a^2 + b^2$

If the equation above is fulfilled, what is the value of $b^6 - 4b^4 + 8b^2$ ?

For more problems, click here .

16 8 4 20

2 solutions

Raj Rajput
Apr 29, 2015

short method put A square = 1 - B square in left hand side equation

squaring both sides keeping square roots on one side and simple on other provides us directly answer = 4

Moderator note:

Can you elaborate on this?

Please elaborate friend?

Aditya Gupta - 3 years, 9 months ago

Taylor Pope
Apr 12, 2015

Since b never appears in an odd power, we can simplify the problem by using x=b² in its place:

1) a + a² + a³ = 1 = a² + x

Then, through simple algebraic manipulation, we can get:

2) a + a² + a³ - 1 = 0

3) 1 - a - a² - a³ = 0

4) x = 1 - a ²

Squaring x:

5) x² = 1 - 2a² + a⁴

Adding and subtracting terms preserves the equality :

6) x² = 1 + (-a + a) + (-a² + a²) + (-a³ + a³) - 2a² + a⁴ + (a - a)

Rearranging :

7) x² = 1 - a - a² - a³ + 2a + a² + a³ + a⁴ - a - 2a²

8) x² = (1 - a - a² - a³) + 2a + a(a + a² + a³ - 1) - 2a²

Applying 2) and 3):

9) x² = (0) + 2a + a(0) - 2a²

10) x² = 2a - 2a²

Now we're ready to look at the question:

11) b^6 - 4b⁴ + 8b² = ?

Substituting in x:

12) x³ - 4x² + 8x = ?

13) x(x² - 4x + 8) = ?

Using 4) and 10):

14) (1 - a²)(2a - 2a² - 4 + 4a² + 8)

15) (1 - a²)(2a + 2a² + 4)

Distribute:

16) 2a + 2a² + 4 - 2a³ - 2a⁴ - 4a²

Simplify:

17) 4 + 2a - 2a²- 2a³ - 2a⁴

18) 4 + 2a(1 - a - a² - a³)

19) 4 + 2a(0) = 4 + 0 = 4

Plug into original question:

20) b^6 - 4b⁴ + 8b² = 4

Exactly how I did it. Beautiful problem!

Aneesh S. - 6 years ago

An algebra problem by Anik Mandal

For more problems, click here .

2 solutions

Moderator note:

0 pending reports