Mister Exam

# Factor -a^4-8*a^2-1 squared

An expression to simplify:

### The solution

You have entered [src]
   4      2
- a  - 8*a  - 1
$$\left(- a^{4} - 8 a^{2}\right) - 1$$
-a^4 - 8*a^2 - 1
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(- a^{4} - 8 a^{2}\right) - 1$$
To do this, let's use the formula
$$a^{5} + a^{2} b + c = a \left(a^{2} + m\right)^{2} + n$$
where
$$m = \frac{b}{2 a}$$
$$n = \frac{4 a c - b^{2}}{4 a}$$
In this case
$$a = -1$$
$$b = -8$$
$$c = -1$$
Then
$$m = 4$$
$$n = 15$$
So,
$$-10$$
General simplification [src]
      4      2
-1 - a  - 8*a 
$$- a^{4} - 8 a^{2} - 1$$
-1 - a^4 - 8*a^2
Factorization [src]
/         ____________\ /         ____________\ /         ____________\ /         ____________\
|        /       ____ | |        /       ____ | |        /       ____ | |        /       ____ |
\a + I*\/  4 - \/ 15  /*\a - I*\/  4 - \/ 15  /*\a + I*\/  4 + \/ 15  /*\a - I*\/  4 + \/ 15  /
$$\left(a - i \sqrt{4 - \sqrt{15}}\right) \left(a + i \sqrt{4 - \sqrt{15}}\right) \left(a + i \sqrt{\sqrt{15} + 4}\right) \left(a - i \sqrt{\sqrt{15} + 4}\right)$$
(((a + i*sqrt(4 - sqrt(15)))*(a - i*sqrt(4 - sqrt(15))))*(a + i*sqrt(4 + sqrt(15))))*(a - i*sqrt(4 + sqrt(15)))
-1.0 - a^4 - 8.0*a^2
-1.0 - a^4 - 8.0*a^2
Combining rational expressions [src]
      2 /      2\
-1 + a *\-8 - a /
$$a^{2} \left(- a^{2} - 8\right) - 1$$
-1 + a^2*(-8 - a^2)
Powers [src]
      4      2
-1 - a  - 8*a 
$$- a^{4} - 8 a^{2} - 1$$
-1 - a^4 - 8*a^2
Common denominator [src]
      4      2
-1 - a  - 8*a 
$$- a^{4} - 8 a^{2} - 1$$
-1 - a^4 - 8*a^2
Trigonometric part [src]
      4      2
-1 - a  - 8*a 
$$- a^{4} - 8 a^{2} - 1$$
-1 - a^4 - 8*a^2
Assemble expression [src]
      4      2
-1 - a  - 8*a 
$$- a^{4} - 8 a^{2} - 1$$
-1 - a^4 - 8*a^2
Rational denominator [src]
      4      2
-1 - a  - 8*a 
$$- a^{4} - 8 a^{2} - 1$$
-1 - a^4 - 8*a^2
Combinatorics [src]
      4      2
-1 - a  - 8*a 
$$- a^{4} - 8 a^{2} - 1$$
-1 - a^4 - 8*a^2
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