Mister Exam
Factor -y^2+7*y*x+11*x^2 squared
The solution
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2 2 - y + 7*y*x + 11*x
$$11 x^{2} + \left(x 7 y - y^{2}\right)$$
-y^2 + (7*y)*x + 11*x^2
General simplification
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2 2 - y + 11*x + 7*x*y
$$11 x^{2} + 7 x y - y^{2}$$
-y^2 + 11*x^2 + 7*x*y
The perfect square
Let's highlight the perfect square of the square three-member
$$11 x^{2} + \left(x 7 y - y^{2}\right)$$
Let us write down the identical expression
$$11 x^{2} + \left(x 7 y - y^{2}\right) = - \frac{93 y^{2}}{44} + \left(11 x^{2} + 7 x y + \frac{49 y^{2}}{44}\right)$$
or
$$11 x^{2} + \left(x 7 y - y^{2}\right) = - \frac{93 y^{2}}{44} + \left(\sqrt{11} x + \frac{7 \sqrt{11} y}{22}\right)^{2}$$
in the view of the product
$$\left(- \sqrt{\frac{93}{44}} y + \left(\sqrt{11} x + \frac{7 \sqrt{11}}{22} y\right)\right) \left(\sqrt{\frac{93}{44}} y + \left(\sqrt{11} x + \frac{7 \sqrt{11}}{22} y\right)\right)$$
$$\left(- \frac{\sqrt{1023}}{22} y + \left(\sqrt{11} x + \frac{7 \sqrt{11}}{22} y\right)\right) \left(\frac{\sqrt{1023}}{22} y + \left(\sqrt{11} x + \frac{7 \sqrt{11}}{22} y\right)\right)$$
$$\left(\sqrt{11} x + y \left(- \frac{\sqrt{1023}}{22} + \frac{7 \sqrt{11}}{22}\right)\right) \left(\sqrt{11} x + y \left(\frac{7 \sqrt{11}}{22} + \frac{\sqrt{1023}}{22}\right)\right)$$
$$\left(\sqrt{11} x + y \left(- \frac{\sqrt{1023}}{22} + \frac{7 \sqrt{11}}{22}\right)\right) \left(\sqrt{11} x + y \left(\frac{7 \sqrt{11}}{22} + \frac{\sqrt{1023}}{22}\right)\right)$$
$$11 x^{2} + \left(x 7 y - y^{2}\right)$$
Let us write down the identical expression
$$11 x^{2} + \left(x 7 y - y^{2}\right) = - \frac{93 y^{2}}{44} + \left(11 x^{2} + 7 x y + \frac{49 y^{2}}{44}\right)$$
or
$$11 x^{2} + \left(x 7 y - y^{2}\right) = - \frac{93 y^{2}}{44} + \left(\sqrt{11} x + \frac{7 \sqrt{11} y}{22}\right)^{2}$$
in the view of the product
$$\left(- \sqrt{\frac{93}{44}} y + \left(\sqrt{11} x + \frac{7 \sqrt{11}}{22} y\right)\right) \left(\sqrt{\frac{93}{44}} y + \left(\sqrt{11} x + \frac{7 \sqrt{11}}{22} y\right)\right)$$
$$\left(- \frac{\sqrt{1023}}{22} y + \left(\sqrt{11} x + \frac{7 \sqrt{11}}{22} y\right)\right) \left(\frac{\sqrt{1023}}{22} y + \left(\sqrt{11} x + \frac{7 \sqrt{11}}{22} y\right)\right)$$
$$\left(\sqrt{11} x + y \left(- \frac{\sqrt{1023}}{22} + \frac{7 \sqrt{11}}{22}\right)\right) \left(\sqrt{11} x + y \left(\frac{7 \sqrt{11}}{22} + \frac{\sqrt{1023}}{22}\right)\right)$$
$$\left(\sqrt{11} x + y \left(- \frac{\sqrt{1023}}{22} + \frac{7 \sqrt{11}}{22}\right)\right) \left(\sqrt{11} x + y \left(\frac{7 \sqrt{11}}{22} + \frac{\sqrt{1023}}{22}\right)\right)$$
Factorization
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/ / ____\\ / / ____\\ | y*\-7 + \/ 93 /| | y*\7 + \/ 93 /| |x - ---------------|*|x + --------------| \ 22 / \ 22 /
$$\left(x - \frac{y \left(-7 + \sqrt{93}\right)}{22}\right) \left(x + \frac{y \left(7 + \sqrt{93}\right)}{22}\right)$$
(x - y*(-7 + sqrt(93))/22)*(x + y*(7 + sqrt(93))/22)
Rational denominator
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2 2 - y + 11*x + 7*x*y
$$11 x^{2} + 7 x y - y^{2}$$
-y^2 + 11*x^2 + 7*x*y
Combining rational expressions
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2 11*x + y*(-y + 7*x)
$$11 x^{2} + y \left(7 x - y\right)$$
11*x^2 + y*(-y + 7*x)