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Expression simplification/ Factor perfect squares/ -8*p^2+6*p+1

Factor -8*p^2+6*p+1 squared

An expression to simplify:

The solution

You have entered [src] 
     2          
- 8*p  + 6*p + 1
$$\left(- 8 p^{2} + 6 p\right) + 1$$ 
-8*p^2 + 6*p + 1
General simplification [src] 
       2      
1 - 8*p  + 6*p
$$- 8 p^{2} + 6 p + 1$$ 
1 - 8*p^2 + 6*p
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(- 8 p^{2} + 6 p\right) + 1$$
To do this, let's use the formula
$$a p^{2} + b p + c = a \left(m + p\right)^{2} + n$$
where
$$m = \frac{b}{2 a}$$
$$n = \frac{4 a c - b^{2}}{4 a}$$
In this case
$$a = -8$$
$$b = 6$$
$$c = 1$$
Then
$$m = - \frac{3}{8}$$
$$n = \frac{17}{8}$$
So,
$$\frac{17}{8} - 8 \left(p - \frac{3}{8}\right)^{2}$$
Factorization [src] 
/            ____\ /            ____\
|      3   \/ 17 | |      3   \/ 17 |
|p + - - + ------|*|p + - - - ------|
\      8     8   / \      8     8   /
$$\left(p + \left(- \frac{3}{8} + \frac{\sqrt{17}}{8}\right)\right) \left(p + \left(- \frac{\sqrt{17}}{8} - \frac{3}{8}\right)\right)$$ 
(p - 3/8 + sqrt(17)/8)*(p - 3/8 - sqrt(17)/8)
Rational denominator [src] 
       2      
1 - 8*p  + 6*p
$$- 8 p^{2} + 6 p + 1$$ 
1 - 8*p^2 + 6*p
Assemble expression [src] 
       2      
1 - 8*p  + 6*p
$$- 8 p^{2} + 6 p + 1$$ 
1 - 8*p^2 + 6*p
Common denominator [src] 
       2      
1 - 8*p  + 6*p
$$- 8 p^{2} + 6 p + 1$$ 
1 - 8*p^2 + 6*p
Numerical answer [src] 
1.0 + 6.0*p - 8.0*p^2
1.0 + 6.0*p - 8.0*p^2
Combining rational expressions [src] 
1 + 2*p*(3 - 4*p)
$$2 p \left(3 - 4 p\right) + 1$$ 
1 + 2*p*(3 - 4*p)
Powers [src] 
       2      
1 - 8*p  + 6*p
$$- 8 p^{2} + 6 p + 1$$ 
1 - 8*p^2 + 6*p
Combinatorics [src] 
       2      
1 - 8*p  + 6*p
$$- 8 p^{2} + 6 p + 1$$ 
1 - 8*p^2 + 6*p
Trigonometric part [src] 
       2      
1 - 8*p  + 6*p
$$- 8 p^{2} + 6 p + 1$$ 
1 - 8*p^2 + 6*p
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