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
General simplification [src]
     2    4
7 + y  - y 
$$- y^{4} + y^{2} + 7$$
7 + y^2 - y^4
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,
$$\frac{29}{4} - \left(y^{2} - \frac{1}{2}\right)^{2}$$
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))
Rational denominator [src]
     2    4
7 + y  - y 
$$- y^{4} + y^{2} + 7$$
7 + y^2 - y^4
Powers [src]
     2    4
7 + y  - y 
$$- y^{4} + y^{2} + 7$$
7 + y^2 - y^4
Assemble expression [src]
     2    4
7 + y  - y 
$$- y^{4} + y^{2} + 7$$
7 + y^2 - y^4
Numerical answer [src]
7.0 + y^2 - y^4
7.0 + y^2 - y^4
Combinatorics [src]
     2    4
7 + y  - y 
$$- y^{4} + y^{2} + 7$$
7 + y^2 - y^4
Trigonometric part [src]
     2    4
7 + y  - y 
$$- y^{4} + y^{2} + 7$$
7 + y^2 - y^4
Combining rational expressions [src]
     2 /     2\
7 + y *\1 - y /
$$y^{2} \left(1 - y^{2}\right) + 7$$
7 + y^2*(1 - y^2)
Common denominator [src]
     2    4
7 + y  - y 
$$- y^{4} + y^{2} + 7$$
7 + y^2 - y^4