Mister Exam
Factor 9*y^2-2*y-2 squared
The solution
Factorization
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/ ____\ / ____\ | 1 \/ 19 | | 1 \/ 19 | |x + - - + ------|*|x + - - - ------| \ 9 9 / \ 9 9 /
$$\left(x + \left(- \frac{1}{9} + \frac{\sqrt{19}}{9}\right)\right) \left(x + \left(- \frac{\sqrt{19}}{9} - \frac{1}{9}\right)\right)$$
(x - 1/9 + sqrt(19)/9)*(x - 1/9 - sqrt(19)/9)
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(9 y^{2} - 2 y\right) - 2$$
To do this, let's use the formula
$$a y^{2} + b y + c = a \left(m + y\right)^{2} + n$$
where
$$m = \frac{b}{2 a}$$
$$n = \frac{4 a c - b^{2}}{4 a}$$
In this case
$$a = 9$$
$$b = -2$$
$$c = -2$$
Then
$$m = - \frac{1}{9}$$
$$n = - \frac{19}{9}$$
So,
$$9 \left(y - \frac{1}{9}\right)^{2} - \frac{19}{9}$$
$$\left(9 y^{2} - 2 y\right) - 2$$
To do this, let's use the formula
$$a y^{2} + b y + c = a \left(m + y\right)^{2} + n$$
where
$$m = \frac{b}{2 a}$$
$$n = \frac{4 a c - b^{2}}{4 a}$$
In this case
$$a = 9$$
$$b = -2$$
$$c = -2$$
Then
$$m = - \frac{1}{9}$$
$$n = - \frac{19}{9}$$
So,
$$9 \left(y - \frac{1}{9}\right)^{2} - \frac{19}{9}$$
Combining rational expressions
[src]
-2 + y*(-2 + 9*y)
$$y \left(9 y - 2\right) - 2$$
-2 + y*(-2 + 9*y)