Mister Exam
Factor y^2+7*y*x+7*x^2 squared
The solution
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2 2 y + 7*y*x + 7*x
$$7 x^{2} + \left(x 7 y + y^{2}\right)$$
y^2 + (7*y)*x + 7*x^2
Factorization
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/ / ____\\ / / ____\\ | y*\-7 + \/ 21 /| | y*\7 + \/ 21 /| |x - ---------------|*|x + --------------| \ 14 / \ 14 /
$$\left(x - \frac{y \left(-7 + \sqrt{21}\right)}{14}\right) \left(x + \frac{y \left(\sqrt{21} + 7\right)}{14}\right)$$
(x - y*(-7 + sqrt(21))/14)*(x + y*(7 + sqrt(21))/14)
The perfect square
Let's highlight the perfect square of the square three-member
$$7 x^{2} + \left(x 7 y + y^{2}\right)$$
Let us write down the identical expression
$$7 x^{2} + \left(x 7 y + y^{2}\right) = - \frac{3 y^{2}}{4} + \left(7 x^{2} + 7 x y + \frac{7 y^{2}}{4}\right)$$
or
$$7 x^{2} + \left(x 7 y + y^{2}\right) = - \frac{3 y^{2}}{4} + \left(\sqrt{7} x + \frac{\sqrt{7} y}{2}\right)^{2}$$
in the view of the product
$$\left(- \sqrt{\frac{3}{4}} y + \left(\sqrt{7} x + \frac{\sqrt{7}}{2} y\right)\right) \left(\sqrt{\frac{3}{4}} y + \left(\sqrt{7} x + \frac{\sqrt{7}}{2} y\right)\right)$$
$$\left(- \frac{\sqrt{3}}{2} y + \left(\sqrt{7} x + \frac{\sqrt{7}}{2} y\right)\right) \left(\frac{\sqrt{3}}{2} y + \left(\sqrt{7} x + \frac{\sqrt{7}}{2} y\right)\right)$$
$$\left(\sqrt{7} x + y \left(- \frac{\sqrt{3}}{2} + \frac{\sqrt{7}}{2}\right)\right) \left(\sqrt{7} x + y \left(\frac{\sqrt{3}}{2} + \frac{\sqrt{7}}{2}\right)\right)$$
$$\left(\sqrt{7} x + y \left(- \frac{\sqrt{3}}{2} + \frac{\sqrt{7}}{2}\right)\right) \left(\sqrt{7} x + y \left(\frac{\sqrt{3}}{2} + \frac{\sqrt{7}}{2}\right)\right)$$
$$7 x^{2} + \left(x 7 y + y^{2}\right)$$
Let us write down the identical expression
$$7 x^{2} + \left(x 7 y + y^{2}\right) = - \frac{3 y^{2}}{4} + \left(7 x^{2} + 7 x y + \frac{7 y^{2}}{4}\right)$$
or
$$7 x^{2} + \left(x 7 y + y^{2}\right) = - \frac{3 y^{2}}{4} + \left(\sqrt{7} x + \frac{\sqrt{7} y}{2}\right)^{2}$$
in the view of the product
$$\left(- \sqrt{\frac{3}{4}} y + \left(\sqrt{7} x + \frac{\sqrt{7}}{2} y\right)\right) \left(\sqrt{\frac{3}{4}} y + \left(\sqrt{7} x + \frac{\sqrt{7}}{2} y\right)\right)$$
$$\left(- \frac{\sqrt{3}}{2} y + \left(\sqrt{7} x + \frac{\sqrt{7}}{2} y\right)\right) \left(\frac{\sqrt{3}}{2} y + \left(\sqrt{7} x + \frac{\sqrt{7}}{2} y\right)\right)$$
$$\left(\sqrt{7} x + y \left(- \frac{\sqrt{3}}{2} + \frac{\sqrt{7}}{2}\right)\right) \left(\sqrt{7} x + y \left(\frac{\sqrt{3}}{2} + \frac{\sqrt{7}}{2}\right)\right)$$
$$\left(\sqrt{7} x + y \left(- \frac{\sqrt{3}}{2} + \frac{\sqrt{7}}{2}\right)\right) \left(\sqrt{7} x + y \left(\frac{\sqrt{3}}{2} + \frac{\sqrt{7}}{2}\right)\right)$$
Combining rational expressions
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2 7*x + y*(y + 7*x)
$$7 x^{2} + y \left(7 x + y\right)$$
7*x^2 + y*(y + 7*x)