Mister Exam
Factor -y^2+y*a+11*a^2 squared
The solution
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2 2 - y + y*a + 11*a
$$11 a^{2} + \left(a y - y^{2}\right)$$
-y^2 + y*a + 11*a^2
Factorization
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/ / ___\\ / / ___\\ | y*\-1 + 3*\/ 5 /| | y*\1 + 3*\/ 5 /| |a - ----------------|*|a + ---------------| \ 22 / \ 22 /
$$\left(a - \frac{y \left(-1 + 3 \sqrt{5}\right)}{22}\right) \left(a + \frac{y \left(1 + 3 \sqrt{5}\right)}{22}\right)$$
(a - y*(-1 + 3*sqrt(5))/22)*(a + y*(1 + 3*sqrt(5))/22)
The perfect square
Let's highlight the perfect square of the square three-member
$$11 a^{2} + \left(a y - y^{2}\right)$$
Let us write down the identical expression
$$11 a^{2} + \left(a y - y^{2}\right) = - \frac{45 y^{2}}{44} + \left(11 a^{2} + a y + \frac{y^{2}}{44}\right)$$
or
$$11 a^{2} + \left(a y - y^{2}\right) = - \frac{45 y^{2}}{44} + \left(\sqrt{11} a + \frac{\sqrt{11} y}{22}\right)^{2}$$
in the view of the product
$$\left(- \sqrt{\frac{45}{44}} y + \left(\sqrt{11} a + \frac{\sqrt{11}}{22} y\right)\right) \left(\sqrt{\frac{45}{44}} y + \left(\sqrt{11} a + \frac{\sqrt{11}}{22} y\right)\right)$$
$$\left(- \frac{3 \sqrt{55}}{22} y + \left(\sqrt{11} a + \frac{\sqrt{11}}{22} y\right)\right) \left(\frac{3 \sqrt{55}}{22} y + \left(\sqrt{11} a + \frac{\sqrt{11}}{22} y\right)\right)$$
$$\left(\sqrt{11} a + y \left(\frac{\sqrt{11}}{22} + \frac{3 \sqrt{55}}{22}\right)\right) \left(\sqrt{11} a + y \left(- \frac{3 \sqrt{55}}{22} + \frac{\sqrt{11}}{22}\right)\right)$$
$$\left(\sqrt{11} a + y \left(\frac{\sqrt{11}}{22} + \frac{3 \sqrt{55}}{22}\right)\right) \left(\sqrt{11} a + y \left(- \frac{3 \sqrt{55}}{22} + \frac{\sqrt{11}}{22}\right)\right)$$
$$11 a^{2} + \left(a y - y^{2}\right)$$
Let us write down the identical expression
$$11 a^{2} + \left(a y - y^{2}\right) = - \frac{45 y^{2}}{44} + \left(11 a^{2} + a y + \frac{y^{2}}{44}\right)$$
or
$$11 a^{2} + \left(a y - y^{2}\right) = - \frac{45 y^{2}}{44} + \left(\sqrt{11} a + \frac{\sqrt{11} y}{22}\right)^{2}$$
in the view of the product
$$\left(- \sqrt{\frac{45}{44}} y + \left(\sqrt{11} a + \frac{\sqrt{11}}{22} y\right)\right) \left(\sqrt{\frac{45}{44}} y + \left(\sqrt{11} a + \frac{\sqrt{11}}{22} y\right)\right)$$
$$\left(- \frac{3 \sqrt{55}}{22} y + \left(\sqrt{11} a + \frac{\sqrt{11}}{22} y\right)\right) \left(\frac{3 \sqrt{55}}{22} y + \left(\sqrt{11} a + \frac{\sqrt{11}}{22} y\right)\right)$$
$$\left(\sqrt{11} a + y \left(\frac{\sqrt{11}}{22} + \frac{3 \sqrt{55}}{22}\right)\right) \left(\sqrt{11} a + y \left(- \frac{3 \sqrt{55}}{22} + \frac{\sqrt{11}}{22}\right)\right)$$
$$\left(\sqrt{11} a + y \left(\frac{\sqrt{11}}{22} + \frac{3 \sqrt{55}}{22}\right)\right) \left(\sqrt{11} a + y \left(- \frac{3 \sqrt{55}}{22} + \frac{\sqrt{11}}{22}\right)\right)$$
Combining rational expressions
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2 11*a + y*(a - y)
$$11 a^{2} + y \left(a - y\right)$$
11*a^2 + y*(a - y)