Mister Exam

# Factor -q^4-4*q^2+5 squared

An expression to simplify:

### The solution

You have entered [src]
   4      2
- q  - 4*q  + 5
$$\left(- q^{4} - 4 q^{2}\right) + 5$$
-q^4 - 4*q^2 + 5
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(- q^{4} - 4 q^{2}\right) + 5$$
To do this, let's use the formula
$$a q^{4} + b q^{2} + c = a \left(m + q^{2}\right)^{2} + n$$
where
$$m = \frac{b}{2 a}$$
$$n = \frac{4 a c - b^{2}}{4 a}$$
In this case
$$a = -1$$
$$b = -4$$
$$c = 5$$
Then
$$m = 2$$
$$n = 9$$
So,
$$9 - \left(q^{2} + 2\right)^{2}$$
General simplification [src]
     4      2
5 - q  - 4*q 
$$- q^{4} - 4 q^{2} + 5$$
5 - q^4 - 4*q^2
Factorization [src]
                /        ___\ /        ___\
(q + 1)*(q - 1)*\q + I*\/ 5 /*\q - I*\/ 5 /
$$\left(q - 1\right) \left(q + 1\right) \left(q + \sqrt{5} i\right) \left(q - \sqrt{5} i\right)$$
(((q + 1)*(q - 1))*(q + i*sqrt(5)))*(q - i*sqrt(5))
Powers [src]
     4      2
5 - q  - 4*q 
$$- q^{4} - 4 q^{2} + 5$$
5 - q^4 - 4*q^2
Assemble expression [src]
     4      2
5 - q  - 4*q 
$$- q^{4} - 4 q^{2} + 5$$
5 - q^4 - 4*q^2
5.0 - q^4 - 4.0*q^2
5.0 - q^4 - 4.0*q^2
Rational denominator [src]
     4      2
5 - q  - 4*q 
$$- q^{4} - 4 q^{2} + 5$$
5 - q^4 - 4*q^2
Combining rational expressions [src]
     2 /      2\
5 + q *\-4 - q /
$$q^{2} \left(- q^{2} - 4\right) + 5$$
5 + q^2*(-4 - q^2)
Trigonometric part [src]
     4      2
5 - q  - 4*q 
$$- q^{4} - 4 q^{2} + 5$$
5 - q^4 - 4*q^2
Combinatorics [src]
                  /     2\
-(1 + q)*(-1 + q)*\5 + q /
$$- \left(q - 1\right) \left(q + 1\right) \left(q^{2} + 5\right)$$
-(1 + q)*(-1 + q)*(5 + q^2)
Common denominator [src]
     4      2
5 - q  - 4*q 
$$- q^{4} - 4 q^{2} + 5$$
5 - q^4 - 4*q^2
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