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Expression simplification/ Factor perfect squares/ y^2-7*y*x+4*x^2

Factor y^2-7*y*x+4*x^2 squared

An expression to simplify:

The solution

You have entered [src] 
 2              2
y  - 7*y*x + 4*x
$$4 x^{2} + \left(- x 7 y + y^{2}\right)$$ 
y^2 - 7*y*x + 4*x^2
General simplification [src] 
 2      2        
y  + 4*x  - 7*x*y
$$4 x^{2} - 7 x y + y^{2}$$ 
y^2 + 4*x^2 - 7*x*y
The perfect square
Let's highlight the perfect square of the square three-member
$$4 x^{2} + \left(- x 7 y + y^{2}\right)$$
Let us write down the identical expression
$$4 x^{2} + \left(- x 7 y + y^{2}\right) = - \frac{33 y^{2}}{16} + \left(4 x^{2} - 7 x y + \frac{49 y^{2}}{16}\right)$$
or
$$4 x^{2} + \left(- x 7 y + y^{2}\right) = - \frac{33 y^{2}}{16} + \left(2 x - \frac{7 y}{4}\right)^{2}$$
in the view of the product
$$\left(- \sqrt{\frac{33}{16}} y + \left(2 x - \frac{7 y}{4}\right)\right) \left(\sqrt{\frac{33}{16}} y + \left(2 x - \frac{7 y}{4}\right)\right)$$
$$\left(- \frac{\sqrt{33}}{4} y + \left(2 x - \frac{7 y}{4}\right)\right) \left(\frac{\sqrt{33}}{4} y + \left(2 x - \frac{7 y}{4}\right)\right)$$
$$\left(2 x + y \left(- \frac{7}{4} - \frac{\sqrt{33}}{4}\right)\right) \left(2 x + y \left(- \frac{7}{4} + \frac{\sqrt{33}}{4}\right)\right)$$
$$\left(2 x + y \left(- \frac{7}{4} - \frac{\sqrt{33}}{4}\right)\right) \left(2 x + y \left(- \frac{7}{4} + \frac{\sqrt{33}}{4}\right)\right)$$
Factorization [src] 
/      /      ____\\ /      /      ____\\
|    y*\7 - \/ 33 /| |    y*\7 + \/ 33 /|
|x - --------------|*|x - --------------|
\          8       / \          8       /
$$\left(x - \frac{y \left(7 - \sqrt{33}\right)}{8}\right) \left(x - \frac{y \left(\sqrt{33} + 7\right)}{8}\right)$$ 
(x - y*(7 - sqrt(33))/8)*(x - y*(7 + sqrt(33))/8)
Numerical answer [src] 
y^2 + 4.0*x^2 - 7.0*x*y
y^2 + 4.0*x^2 - 7.0*x*y
Assemble expression [src] 
 2      2        
y  + 4*x  - 7*x*y
$$4 x^{2} - 7 x y + y^{2}$$ 
y^2 + 4*x^2 - 7*x*y
Rational denominator [src] 
 2      2        
y  + 4*x  - 7*x*y
$$4 x^{2} - 7 x y + y^{2}$$ 
y^2 + 4*x^2 - 7*x*y
Trigonometric part [src] 
 2      2        
y  + 4*x  - 7*x*y
$$4 x^{2} - 7 x y + y^{2}$$ 
y^2 + 4*x^2 - 7*x*y
Combining rational expressions [src] 
   2              
4*x  + y*(y - 7*x)
$$4 x^{2} + y \left(- 7 x + y\right)$$ 
4*x^2 + y*(y - 7*x)
Powers [src] 
 2      2        
y  + 4*x  - 7*x*y
$$4 x^{2} - 7 x y + y^{2}$$ 
y^2 + 4*x^2 - 7*x*y
Common denominator [src] 
 2      2        
y  + 4*x  - 7*x*y
$$4 x^{2} - 7 x y + y^{2}$$ 
y^2 + 4*x^2 - 7*x*y
Combinatorics [src] 
 2      2        
y  + 4*x  - 7*x*y
$$4 x^{2} - 7 x y + y^{2}$$ 
y^2 + 4*x^2 - 7*x*y
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