Mister Exam
Factor -x^2+9*x*y+6*y^2 squared
The solution
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2 2 - x + 9*x*y + 6*y
$$6 y^{2} + \left(- x^{2} + 9 x y\right)$$
-x^2 + (9*x)*y + 6*y^2
The perfect square
Let's highlight the perfect square of the square three-member
$$6 y^{2} + \left(- x^{2} + 9 x y\right)$$
Let us write down the identical expression
$$6 y^{2} + \left(- x^{2} + 9 x y\right) = \frac{105 y^{2}}{4} + \left(- x^{2} + 9 x y - \frac{81 y^{2}}{4}\right)$$
or
$$6 y^{2} + \left(- x^{2} + 9 x y\right) = \frac{105 y^{2}}{4} - \left(x - \frac{9 y}{2}\right)^{2}$$
$$6 y^{2} + \left(- x^{2} + 9 x y\right)$$
Let us write down the identical expression
$$6 y^{2} + \left(- x^{2} + 9 x y\right) = \frac{105 y^{2}}{4} + \left(- x^{2} + 9 x y - \frac{81 y^{2}}{4}\right)$$
or
$$6 y^{2} + \left(- x^{2} + 9 x y\right) = \frac{105 y^{2}}{4} - \left(x - \frac{9 y}{2}\right)^{2}$$
Factorization
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/ / _____\\ / / _____\\ | y*\9 - \/ 105 /| | y*\9 + \/ 105 /| |x - ---------------|*|x - ---------------| \ 2 / \ 2 /
$$\left(x - \frac{y \left(9 - \sqrt{105}\right)}{2}\right) \left(x - \frac{y \left(9 + \sqrt{105}\right)}{2}\right)$$
(x - y*(9 - sqrt(105))/2)*(x - y*(9 + sqrt(105))/2)
General simplification
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2 2 - x + 6*y + 9*x*y
$$- x^{2} + 9 x y + 6 y^{2}$$
-x^2 + 6*y^2 + 9*x*y
Combining rational expressions
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2 6*y + x*(-x + 9*y)
$$x \left(- x + 9 y\right) + 6 y^{2}$$
6*y^2 + x*(-x + 9*y)