Mister Exam
Factor x^2-12*x*y+4*y^2 squared
The solution
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2 2 x - 12*x*y + 4*y
$$4 y^{2} + \left(x^{2} - 12 x y\right)$$
x^2 - 12*x*y + 4*y^2
The perfect square
Let's highlight the perfect square of the square three-member
$$4 y^{2} + \left(x^{2} - 12 x y\right)$$
Let us write down the identical expression
$$4 y^{2} + \left(x^{2} - 12 x y\right) = - 32 y^{2} + \left(x^{2} - 12 x y + 36 y^{2}\right)$$
or
$$4 y^{2} + \left(x^{2} - 12 x y\right) = - 32 y^{2} + \left(x - 6 y\right)^{2}$$
in the view of the product
$$\left(- \sqrt{32} y + \left(x - 6 y\right)\right) \left(\sqrt{32} y + \left(x - 6 y\right)\right)$$
$$\left(- 4 \sqrt{2} y + \left(x - 6 y\right)\right) \left(4 \sqrt{2} y + \left(x - 6 y\right)\right)$$
$$\left(x + y \left(-6 - 4 \sqrt{2}\right)\right) \left(x + y \left(-6 + 4 \sqrt{2}\right)\right)$$
$$\left(x + y \left(-6 - 4 \sqrt{2}\right)\right) \left(x + y \left(-6 + 4 \sqrt{2}\right)\right)$$
$$4 y^{2} + \left(x^{2} - 12 x y\right)$$
Let us write down the identical expression
$$4 y^{2} + \left(x^{2} - 12 x y\right) = - 32 y^{2} + \left(x^{2} - 12 x y + 36 y^{2}\right)$$
or
$$4 y^{2} + \left(x^{2} - 12 x y\right) = - 32 y^{2} + \left(x - 6 y\right)^{2}$$
in the view of the product
$$\left(- \sqrt{32} y + \left(x - 6 y\right)\right) \left(\sqrt{32} y + \left(x - 6 y\right)\right)$$
$$\left(- 4 \sqrt{2} y + \left(x - 6 y\right)\right) \left(4 \sqrt{2} y + \left(x - 6 y\right)\right)$$
$$\left(x + y \left(-6 - 4 \sqrt{2}\right)\right) \left(x + y \left(-6 + 4 \sqrt{2}\right)\right)$$
$$\left(x + y \left(-6 - 4 \sqrt{2}\right)\right) \left(x + y \left(-6 + 4 \sqrt{2}\right)\right)$$
Factorization
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/ / ___\\ / / ___\\ \x - 2*y*\3 - 2*\/ 2 //*\x - 2*y*\3 + 2*\/ 2 //
$$\left(x - 2 y \left(3 - 2 \sqrt{2}\right)\right) \left(x - 2 y \left(2 \sqrt{2} + 3\right)\right)$$
(x - 2*y*(3 - 2*sqrt(2)))*(x - 2*y*(3 + 2*sqrt(2)))
Combining rational expressions
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2 4*y + x*(x - 12*y)
$$x \left(x - 12 y\right) + 4 y^{2}$$
4*y^2 + x*(x - 12*y)