Mister Exam

# Factor y^4-y^2-7 squared

An expression to simplify:

### The solution

You have entered [src]
 4    2
y  - y  - 7
$$\left(y^{4} - y^{2}\right) - 7$$
y^4 - y^2 - 7
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(y^{4} - y^{2}\right) - 7$$
To do this, let's use the formula
$$a y^{4} + b y^{2} + c = a \left(m + y^{2}\right)^{2} + n$$
where
$$m = \frac{b}{2 a}$$
$$n = \frac{4 a c - b^{2}}{4 a}$$
In this case
$$a = 1$$
$$b = -1$$
$$c = -7$$
Then
$$m = - \frac{1}{2}$$
$$n = - \frac{29}{4}$$
So,
$$\left(y^{2} - \frac{1}{2}\right)^{2} - \frac{29}{4}$$
General simplification [src]
      4    2
-7 + y  - y 
$$y^{4} - y^{2} - 7$$
-7 + y^4 - y^2
Factorization [src]
/           ______________\ /           ______________\ /         ____________\ /         ____________\
|          /         ____ | |          /         ____ | |        /       ____ | |        /       ____ |
|         /    1   \/ 29  | |         /    1   \/ 29  | |       /  1   \/ 29  | |       /  1   \/ 29  |
|x + I*  /   - - + ------ |*|x - I*  /   - - + ------ |*|x +   /   - + ------ |*|x -   /   - + ------ |
\      \/      2     2    / \      \/      2     2    / \    \/    2     2    / \    \/    2     2    /
$$\left(x - i \sqrt{- \frac{1}{2} + \frac{\sqrt{29}}{2}}\right) \left(x + i \sqrt{- \frac{1}{2} + \frac{\sqrt{29}}{2}}\right) \left(x + \sqrt{\frac{1}{2} + \frac{\sqrt{29}}{2}}\right) \left(x - \sqrt{\frac{1}{2} + \frac{\sqrt{29}}{2}}\right)$$
(((x + i*sqrt(-1/2 + sqrt(29)/2))*(x - i*sqrt(-1/2 + sqrt(29)/2)))*(x + sqrt(1/2 + sqrt(29)/2)))*(x - sqrt(1/2 + sqrt(29)/2))
Common denominator [src]
      4    2
-7 + y  - y 
$$y^{4} - y^{2} - 7$$
-7 + y^4 - y^2
Rational denominator [src]
      4    2
-7 + y  - y 
$$y^{4} - y^{2} - 7$$
-7 + y^4 - y^2
Assemble expression [src]
      4    2
-7 + y  - y 
$$y^{4} - y^{2} - 7$$
-7 + y^4 - y^2
Trigonometric part [src]
      4    2
-7 + y  - y 
$$y^{4} - y^{2} - 7$$
-7 + y^4 - y^2
-7.0 + y^4 - y^2
-7.0 + y^4 - y^2
Combining rational expressions [src]
      2 /      2\
-7 + y *\-1 + y /
$$y^{2} \left(y^{2} - 1\right) - 7$$
-7 + y^2*(-1 + y^2)
Combinatorics [src]
      4    2
-7 + y  - y 
$$y^{4} - y^{2} - 7$$
-7 + y^4 - y^2
Powers [src]
      4    2
-7 + y  - y 
$$y^{4} - y^{2} - 7$$
-7 + y^4 - y^2