Mister Exam
Factor -8*y^2+y+12 squared
The solution
Factorization
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/ _____\ / _____\ | 1 \/ 385 | | 1 \/ 385 | |x + - -- + -------|*|x + - -- - -------| \ 16 16 / \ 16 16 /
$$\left(x + \left(- \frac{1}{16} + \frac{\sqrt{385}}{16}\right)\right) \left(x + \left(- \frac{\sqrt{385}}{16} - \frac{1}{16}\right)\right)$$
(x - 1/16 + sqrt(385)/16)*(x - 1/16 - sqrt(385)/16)
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(- 8 y^{2} + y\right) + 12$$
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
$$a = -8$$
$$b = 1$$
$$c = 12$$
Then
$$m = - \frac{1}{16}$$
$$n = \frac{385}{32}$$
So,
$$\frac{385}{32} - 8 \left(y - \frac{1}{16}\right)^{2}$$
$$\left(- 8 y^{2} + y\right) + 12$$
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
$$a = -8$$
$$b = 1$$
$$c = 12$$
Then
$$m = - \frac{1}{16}$$
$$n = \frac{385}{32}$$
So,
$$\frac{385}{32} - 8 \left(y - \frac{1}{16}\right)^{2}$$
Combining rational expressions
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12 + y*(1 - 8*y)
$$y \left(1 - 8 y\right) + 12$$
12 + y*(1 - 8*y)