Mister Exam
Factor 4*b^2-b-2 squared
The solution
Factorization
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/ ____\ / ____\ | 1 \/ 33 | | 1 \/ 33 | |b + - - + ------|*|b + - - - ------| \ 8 8 / \ 8 8 /
$$\left(b + \left(- \frac{1}{8} + \frac{\sqrt{33}}{8}\right)\right) \left(b + \left(- \frac{\sqrt{33}}{8} - \frac{1}{8}\right)\right)$$
(b - 1/8 + sqrt(33)/8)*(b - 1/8 - sqrt(33)/8)
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(4 b^{2} - b\right) - 2$$
To do this, let's use the formula
$$a b^{2} + b^{2} + c = a \left(b + m\right)^{2} + n$$
where
$$m = \frac{b}{2 a}$$
$$n = \frac{4 a c - b^{2}}{4 a}$$
In this case
$$a = 4$$
$$b = -1$$
$$c = -2$$
Then
$$m = - \frac{1}{8}$$
$$n = - \frac{33}{16}$$
So,
$$4 \left(b - \frac{1}{8}\right)^{2} - \frac{33}{16}$$
$$\left(4 b^{2} - b\right) - 2$$
To do this, let's use the formula
$$a b^{2} + b^{2} + c = a \left(b + m\right)^{2} + n$$
where
$$m = \frac{b}{2 a}$$
$$n = \frac{4 a c - b^{2}}{4 a}$$
In this case
$$a = 4$$
$$b = -1$$
$$c = -2$$
Then
$$m = - \frac{1}{8}$$
$$n = - \frac{33}{16}$$
So,
$$4 \left(b - \frac{1}{8}\right)^{2} - \frac{33}{16}$$
Combining rational expressions
[src]
-2 + b*(-1 + 4*b)
$$b \left(4 b - 1\right) - 2$$
-2 + b*(-1 + 4*b)