Mister Exam
Factor y^2+7*y*x+x^2 squared
The solution
The perfect square
Let's highlight the perfect square of the square three-member
$$x^{2} + \left(x 7 y + y^{2}\right)$$
Let us write down the identical expression
$$x^{2} + \left(x 7 y + y^{2}\right) = - \frac{45 y^{2}}{4} + \left(x^{2} + 7 x y + \frac{49 y^{2}}{4}\right)$$
or
$$x^{2} + \left(x 7 y + y^{2}\right) = - \frac{45 y^{2}}{4} + \left(x + \frac{7 y}{2}\right)^{2}$$
in the view of the product
$$\left(- \sqrt{\frac{45}{4}} y + \left(x + \frac{7 y}{2}\right)\right) \left(\sqrt{\frac{45}{4}} y + \left(x + \frac{7 y}{2}\right)\right)$$
$$\left(- \frac{3 \sqrt{5}}{2} y + \left(x + \frac{7 y}{2}\right)\right) \left(\frac{3 \sqrt{5}}{2} y + \left(x + \frac{7 y}{2}\right)\right)$$
$$\left(x + y \left(\frac{7}{2} - \frac{3 \sqrt{5}}{2}\right)\right) \left(x + y \left(\frac{3 \sqrt{5}}{2} + \frac{7}{2}\right)\right)$$
$$\left(x + y \left(\frac{7}{2} - \frac{3 \sqrt{5}}{2}\right)\right) \left(x + y \left(\frac{3 \sqrt{5}}{2} + \frac{7}{2}\right)\right)$$
$$x^{2} + \left(x 7 y + y^{2}\right)$$
Let us write down the identical expression
$$x^{2} + \left(x 7 y + y^{2}\right) = - \frac{45 y^{2}}{4} + \left(x^{2} + 7 x y + \frac{49 y^{2}}{4}\right)$$
or
$$x^{2} + \left(x 7 y + y^{2}\right) = - \frac{45 y^{2}}{4} + \left(x + \frac{7 y}{2}\right)^{2}$$
in the view of the product
$$\left(- \sqrt{\frac{45}{4}} y + \left(x + \frac{7 y}{2}\right)\right) \left(\sqrt{\frac{45}{4}} y + \left(x + \frac{7 y}{2}\right)\right)$$
$$\left(- \frac{3 \sqrt{5}}{2} y + \left(x + \frac{7 y}{2}\right)\right) \left(\frac{3 \sqrt{5}}{2} y + \left(x + \frac{7 y}{2}\right)\right)$$
$$\left(x + y \left(\frac{7}{2} - \frac{3 \sqrt{5}}{2}\right)\right) \left(x + y \left(\frac{3 \sqrt{5}}{2} + \frac{7}{2}\right)\right)$$
$$\left(x + y \left(\frac{7}{2} - \frac{3 \sqrt{5}}{2}\right)\right) \left(x + y \left(\frac{3 \sqrt{5}}{2} + \frac{7}{2}\right)\right)$$
Factorization
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/ / ___\\ / / ___\\ | y*\-7 + 3*\/ 5 /| | y*\7 + 3*\/ 5 /| |x - ----------------|*|x + ---------------| \ 2 / \ 2 /
$$\left(x - \frac{y \left(-7 + 3 \sqrt{5}\right)}{2}\right) \left(x + \frac{y \left(3 \sqrt{5} + 7\right)}{2}\right)$$
(x - y*(-7 + 3*sqrt(5))/2)*(x + y*(7 + 3*sqrt(5))/2)
Combining rational expressions
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2 x + y*(y + 7*x)
$$x^{2} + y \left(7 x + y\right)$$
x^2 + y*(y + 7*x)