Mister Exam

Other calculators

Expression simplification/ Factor perfect squares/ y^4+2*y^2+3

Factor y^4+2*y^2+3 squared

An expression to simplify:

The solution

You have entered [src] 
 4      2    
y  + 2*y  + 3
$$\left(y^{4} + 2 y^{2}\right) + 3$$ 
y^4 + 2*y^2 + 3
Factorization [src] 
/             /    /  ___\\              /    /  ___\\\ /             /    /  ___\\              /    /  ___\\\ /               /    /  ___\\              /    /  ___\\\ /               /    /  ___\\              /    /  ___\\\
|    4 ___    |atan\\/ 2 /|     4 ___    |atan\\/ 2 /|| |    4 ___    |atan\\/ 2 /|     4 ___    |atan\\/ 2 /|| |      4 ___    |atan\\/ 2 /|     4 ___    |atan\\/ 2 /|| |      4 ___    |atan\\/ 2 /|     4 ___    |atan\\/ 2 /||
|x + \/ 3 *sin|-----------| + I*\/ 3 *cos|-----------||*|x + \/ 3 *sin|-----------| - I*\/ 3 *cos|-----------||*|x + - \/ 3 *sin|-----------| + I*\/ 3 *cos|-----------||*|x + - \/ 3 *sin|-----------| - I*\/ 3 *cos|-----------||
\             \     2     /              \     2     // \             \     2     /              \     2     // \               \     2     /              \     2     // \               \     2     /              \     2     //
$$\left(x + \left(\sqrt[4]{3} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{2} \right)}}{2} \right)} - \sqrt[4]{3} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{2} \right)}}{2} \right)}\right)\right) \left(x + \left(\sqrt[4]{3} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{2} \right)}}{2} \right)} + \sqrt[4]{3} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{2} \right)}}{2} \right)}\right)\right) \left(x + \left(- \sqrt[4]{3} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{2} \right)}}{2} \right)} + \sqrt[4]{3} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{2} \right)}}{2} \right)}\right)\right) \left(x + \left(- \sqrt[4]{3} \sin{\left(\frac{\operatorname{atan}{\left(\sqrt{2} \right)}}{2} \right)} - \sqrt[4]{3} i \cos{\left(\frac{\operatorname{atan}{\left(\sqrt{2} \right)}}{2} \right)}\right)\right)$$ 
(((x + 3^(1/4)*sin(atan(sqrt(2))/2) + i*3^(1/4)*cos(atan(sqrt(2))/2))*(x + 3^(1/4)*sin(atan(sqrt(2))/2) - i*3^(1/4)*cos(atan(sqrt(2))/2)))*(x - 3^(1/4)*sin(atan(sqrt(2))/2) + i*3^(1/4)*cos(atan(sqrt(2))/2)))*(x - 3^(1/4)*sin(atan(sqrt(2))/2) - i*3^(1/4)*cos(atan(sqrt(2))/2))
General simplification [src] 
     4      2
3 + y  + 2*y
$$y^{4} + 2 y^{2} + 3$$ 
3 + y^4 + 2*y^2
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(y^{4} + 2 y^{2}\right) + 3$$
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
$$a = 1$$
$$b = 2$$
$$c = 3$$
Then
$$m = 1$$
$$n = 2$$
So,
$$\left(y^{2} + 1\right)^{2} + 2$$
Numerical answer [src] 
3.0 + y^4 + 2.0*y^2
3.0 + y^4 + 2.0*y^2
Combining rational expressions [src] 
     2 /     2\
3 + y *\2 + y /
$$y^{2} \left(y^{2} + 2\right) + 3$$ 
3 + y^2*(2 + y^2)
Trigonometric part [src] 
     4      2
3 + y  + 2*y
$$y^{4} + 2 y^{2} + 3$$ 
3 + y^4 + 2*y^2
Combinatorics [src] 
     4      2
3 + y  + 2*y
$$y^{4} + 2 y^{2} + 3$$ 
3 + y^4 + 2*y^2
Powers [src] 
     4      2
3 + y  + 2*y
$$y^{4} + 2 y^{2} + 3$$ 
3 + y^4 + 2*y^2
Common denominator [src] 
     4      2
3 + y  + 2*y
$$y^{4} + 2 y^{2} + 3$$ 
3 + y^4 + 2*y^2
Rational denominator [src] 
     4      2
3 + y  + 2*y
$$y^{4} + 2 y^{2} + 3$$ 
3 + y^4 + 2*y^2
Assemble expression [src] 
     4      2
3 + y  + 2*y
$$y^{4} + 2 y^{2} + 3$$ 
3 + y^4 + 2*y^2
    © Mister Exam – Calculator