Mister Exam
Factor -y^2-10*y*p-2*p^2 squared
The solution
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2 2 - y - 10*y*p - 2*p
$$- 2 p^{2} + \left(- p 10 y - y^{2}\right)$$
-y^2 - 10*y*p - 2*p^2
The perfect square
Let's highlight the perfect square of the square three-member
$$- 2 p^{2} + \left(- p 10 y - y^{2}\right)$$
Let us write down the identical expression
$$- 2 p^{2} + \left(- p 10 y - y^{2}\right) = \frac{23 y^{2}}{2} + \left(- 2 p^{2} - 10 p y - \frac{25 y^{2}}{2}\right)$$
or
$$- 2 p^{2} + \left(- p 10 y - y^{2}\right) = \frac{23 y^{2}}{2} - \left(\sqrt{2} p + \frac{5 \sqrt{2} y}{2}\right)^{2}$$
$$- 2 p^{2} + \left(- p 10 y - y^{2}\right)$$
Let us write down the identical expression
$$- 2 p^{2} + \left(- p 10 y - y^{2}\right) = \frac{23 y^{2}}{2} + \left(- 2 p^{2} - 10 p y - \frac{25 y^{2}}{2}\right)$$
or
$$- 2 p^{2} + \left(- p 10 y - y^{2}\right) = \frac{23 y^{2}}{2} - \left(\sqrt{2} p + \frac{5 \sqrt{2} y}{2}\right)^{2}$$
General simplification
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2 2 - y - 2*p - 10*p*y
$$- 2 p^{2} - 10 p y - y^{2}$$
-y^2 - 2*p^2 - 10*p*y
Factorization
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/ / ____\\ / / ____\\ | y*\-5 + \/ 23 /| | y*\5 + \/ 23 /| |p - ---------------|*|p + --------------| \ 2 / \ 2 /
$$\left(p - \frac{y \left(-5 + \sqrt{23}\right)}{2}\right) \left(p + \frac{y \left(\sqrt{23} + 5\right)}{2}\right)$$
(p - y*(-5 + sqrt(23))/2)*(p + y*(5 + sqrt(23))/2)
Trigonometric part
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2 2 - y - 2*p - 10*p*y
$$- 2 p^{2} - 10 p y - y^{2}$$
-y^2 - 2*p^2 - 10*p*y
Assemble expression
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2 2 - y - 2*p - 10*p*y
$$- 2 p^{2} - 10 p y - y^{2}$$
-y^2 - 2*p^2 - 10*p*y
Common denominator
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2 2 - y - 2*p - 10*p*y
$$- 2 p^{2} - 10 p y - y^{2}$$
-y^2 - 2*p^2 - 10*p*y
Combining rational expressions
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2 - 2*p + y*(-y - 10*p)
$$- 2 p^{2} + y \left(- 10 p - y\right)$$
-2*p^2 + y*(-y - 10*p)
Rational denominator
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2 2 - y - 2*p - 10*p*y
$$- 2 p^{2} - 10 p y - y^{2}$$
-y^2 - 2*p^2 - 10*p*y