Mister Exam

Other calculators

Expression simplification/ Factorization polynomials/ z^2-z+5

Factor polynomial z^2-z+5

An expression to simplify:

The solution

You have entered [src] 
 2        
z  - z + 5
(z2−z)+5\left(z^{2} - z\right) + 5 
z^2 - z + 5
General simplification [src] 
     2    
5 + z  - z
z2−z+5z^{2} - z + 5 
5 + z^2 - z
Factorization [src] 
/              ____\ /              ____\
|      1   I*\/ 19 | |      1   I*\/ 19 |
|x + - - + --------|*|x + - - - --------|
\      2      2    / \      2      2    /
(x+(−12−19i2))(x+(−12+19i2))\left(x + \left(- \frac{1}{2} - \frac{\sqrt{19} i}{2}\right)\right) \left(x + \left(- \frac{1}{2} + \frac{\sqrt{19} i}{2}\right)\right) 
(x - 1/2 + i*sqrt(19)/2)*(x - 1/2 - i*sqrt(19)/2)
The perfect square
Let's highlight the perfect square of the square three-member
(z2−z)+5\left(z^{2} - z\right) + 5
To do this, let's use the formula
az2+bz+c=a(m+z)2+na z^{2} + b z + c = a \left(m + z\right)^{2} + n
where
m=b2am = \frac{b}{2 a}
n=4ac−b24an = \frac{4 a c - b^{2}}{4 a}
In this case
a=1a = 1
b=−1b = -1
c=5c = 5
Then
m=−12m = - \frac{1}{2}
n=194n = \frac{19}{4}
So,
(z−12)2+194\left(z - \frac{1}{2}\right)^{2} + \frac{19}{4}
Combinatorics [src] 
     2    
5 + z  - z
z2−z+5z^{2} - z + 5 
5 + z^2 - z
Powers [src] 
     2    
5 + z  - z
z2−z+5z^{2} - z + 5 
5 + z^2 - z
Rational denominator [src] 
     2    
5 + z  - z
z2−z+5z^{2} - z + 5 
5 + z^2 - z
Common denominator [src] 
     2    
5 + z  - z
z2−z+5z^{2} - z + 5 
5 + z^2 - z
Trigonometric part [src] 
     2    
5 + z  - z
z2−z+5z^{2} - z + 5 
5 + z^2 - z
Assemble expression [src] 
     2    
5 + z  - z
z2−z+5z^{2} - z + 5 
5 + z^2 - z
Numerical answer [src] 
5.0 + z^2 - z
5.0 + z^2 - z
Combining rational expressions [src] 
5 + z*(-1 + z)
z(z−1)+5z \left(z - 1\right) + 5 
5 + z*(-1 + z)
    © Mister Exam – Calculator