Mister Exam
Factor -5*b^2-2*b+1 squared
The solution
Factorization
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/ ___\ / ___\ | 1 \/ 6 | | 1 \/ 6 | |b + - - -----|*|b + - + -----| \ 5 5 / \ 5 5 /
$$\left(b + \left(\frac{1}{5} - \frac{\sqrt{6}}{5}\right)\right) \left(b + \left(\frac{1}{5} + \frac{\sqrt{6}}{5}\right)\right)$$
(b + 1/5 - sqrt(6)/5)*(b + 1/5 + sqrt(6)/5)
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(- 5 b^{2} - 2 b\right) + 1$$
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
$$a = -5$$
$$b = -2$$
$$c = 1$$
Then
$$m = \frac{1}{5}$$
$$n = \frac{6}{5}$$
So,
$$\frac{6}{5} - 5 \left(b + \frac{1}{5}\right)^{2}$$
$$\left(- 5 b^{2} - 2 b\right) + 1$$
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
$$a = -5$$
$$b = -2$$
$$c = 1$$
Then
$$m = \frac{1}{5}$$
$$n = \frac{6}{5}$$
So,
$$\frac{6}{5} - 5 \left(b + \frac{1}{5}\right)^{2}$$
Combining rational expressions
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1 + b*(-2 - 5*b)
$$b \left(- 5 b - 2\right) + 1$$
1 + b*(-2 - 5*b)