Mister Exam
Factor -6*y^2-2*y*x+13*x^2 squared
The solution
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2 2 - 6*y - 2*y*x + 13*x
$$13 x^{2} + \left(- x 2 y - 6 y^{2}\right)$$
-6*y^2 - 2*y*x + 13*x^2
The perfect square
Let's highlight the perfect square of the square three-member
$$13 x^{2} + \left(- x 2 y - 6 y^{2}\right)$$
Let us write down the identical expression
$$13 x^{2} + \left(- x 2 y - 6 y^{2}\right) = - \frac{79 y^{2}}{13} + \left(13 x^{2} - 2 x y + \frac{y^{2}}{13}\right)$$
or
$$13 x^{2} + \left(- x 2 y - 6 y^{2}\right) = - \frac{79 y^{2}}{13} + \left(\sqrt{13} x - \frac{\sqrt{13} y}{13}\right)^{2}$$
in the view of the product
$$\left(- \sqrt{\frac{79}{13}} y + \left(\sqrt{13} x + - \frac{\sqrt{13}}{13} y\right)\right) \left(\sqrt{\frac{79}{13}} y + \left(\sqrt{13} x + - \frac{\sqrt{13}}{13} y\right)\right)$$
$$\left(- \frac{\sqrt{1027}}{13} y + \left(\sqrt{13} x + - \frac{\sqrt{13}}{13} y\right)\right) \left(\frac{\sqrt{1027}}{13} y + \left(\sqrt{13} x + - \frac{\sqrt{13}}{13} y\right)\right)$$
$$\left(\sqrt{13} x + y \left(- \frac{\sqrt{1027}}{13} - \frac{\sqrt{13}}{13}\right)\right) \left(\sqrt{13} x + y \left(- \frac{\sqrt{13}}{13} + \frac{\sqrt{1027}}{13}\right)\right)$$
$$\left(\sqrt{13} x + y \left(- \frac{\sqrt{1027}}{13} - \frac{\sqrt{13}}{13}\right)\right) \left(\sqrt{13} x + y \left(- \frac{\sqrt{13}}{13} + \frac{\sqrt{1027}}{13}\right)\right)$$
$$13 x^{2} + \left(- x 2 y - 6 y^{2}\right)$$
Let us write down the identical expression
$$13 x^{2} + \left(- x 2 y - 6 y^{2}\right) = - \frac{79 y^{2}}{13} + \left(13 x^{2} - 2 x y + \frac{y^{2}}{13}\right)$$
or
$$13 x^{2} + \left(- x 2 y - 6 y^{2}\right) = - \frac{79 y^{2}}{13} + \left(\sqrt{13} x - \frac{\sqrt{13} y}{13}\right)^{2}$$
in the view of the product
$$\left(- \sqrt{\frac{79}{13}} y + \left(\sqrt{13} x + - \frac{\sqrt{13}}{13} y\right)\right) \left(\sqrt{\frac{79}{13}} y + \left(\sqrt{13} x + - \frac{\sqrt{13}}{13} y\right)\right)$$
$$\left(- \frac{\sqrt{1027}}{13} y + \left(\sqrt{13} x + - \frac{\sqrt{13}}{13} y\right)\right) \left(\frac{\sqrt{1027}}{13} y + \left(\sqrt{13} x + - \frac{\sqrt{13}}{13} y\right)\right)$$
$$\left(\sqrt{13} x + y \left(- \frac{\sqrt{1027}}{13} - \frac{\sqrt{13}}{13}\right)\right) \left(\sqrt{13} x + y \left(- \frac{\sqrt{13}}{13} + \frac{\sqrt{1027}}{13}\right)\right)$$
$$\left(\sqrt{13} x + y \left(- \frac{\sqrt{1027}}{13} - \frac{\sqrt{13}}{13}\right)\right) \left(\sqrt{13} x + y \left(- \frac{\sqrt{13}}{13} + \frac{\sqrt{1027}}{13}\right)\right)$$
Factorization
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/ / ____\\ / / ____\\ | y*\1 - \/ 79 /| | y*\1 + \/ 79 /| |x - --------------|*|x - --------------| \ 13 / \ 13 /
$$\left(x - \frac{y \left(1 - \sqrt{79}\right)}{13}\right) \left(x - \frac{y \left(1 + \sqrt{79}\right)}{13}\right)$$
(x - y*(1 - sqrt(79))/13)*(x - y*(1 + sqrt(79))/13)
General simplification
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2 2 - 6*y + 13*x - 2*x*y
$$13 x^{2} - 2 x y - 6 y^{2}$$
-6*y^2 + 13*x^2 - 2*x*y
Assemble expression
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2 2 - 6*y + 13*x - 2*x*y
$$13 x^{2} - 2 x y - 6 y^{2}$$
-6*y^2 + 13*x^2 - 2*x*y
Combining rational expressions
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2 13*x + 2*y*(-x - 3*y)
$$13 x^{2} + 2 y \left(- x - 3 y\right)$$
13*x^2 + 2*y*(-x - 3*y)
Rational denominator
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2 2 - 6*y + 13*x - 2*x*y
$$13 x^{2} - 2 x y - 6 y^{2}$$
-6*y^2 + 13*x^2 - 2*x*y
Trigonometric part
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2 2 - 6*y + 13*x - 2*x*y
$$13 x^{2} - 2 x y - 6 y^{2}$$
-6*y^2 + 13*x^2 - 2*x*y
Common denominator
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2 2 - 6*y + 13*x - 2*x*y
$$13 x^{2} - 2 x y - 6 y^{2}$$
-6*y^2 + 13*x^2 - 2*x*y