Mister Exam
Factor a^2-a-13 squared
The solution
Factorization
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/ ____\ / ____\ | 1 \/ 53 | | 1 \/ 53 | |a + - - + ------|*|a + - - - ------| \ 2 2 / \ 2 2 /
$$\left(a + \left(- \frac{1}{2} + \frac{\sqrt{53}}{2}\right)\right) \left(a + \left(- \frac{\sqrt{53}}{2} - \frac{1}{2}\right)\right)$$
(a - 1/2 + sqrt(53)/2)*(a - 1/2 - sqrt(53)/2)
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(a^{2} - a\right) - 13$$
To do this, let's use the formula
$$a^{3} + a b + c = a \left(a + m\right)^{2} + n$$
where
$$m = \frac{b}{2 a}$$
$$n = \frac{4 a c - b^{2}}{4 a}$$
In this case
$$a = 1$$
$$b = -1$$
$$c = -13$$
Then
$$m = - \frac{1}{2}$$
$$n = - \frac{53}{4}$$
So,
$$-13$$
$$\left(a^{2} - a\right) - 13$$
To do this, let's use the formula
$$a^{3} + a b + c = a \left(a + m\right)^{2} + n$$
where
$$m = \frac{b}{2 a}$$
$$n = \frac{4 a c - b^{2}}{4 a}$$
In this case
$$a = 1$$
$$b = -1$$
$$c = -13$$
Then
$$m = - \frac{1}{2}$$
$$n = - \frac{53}{4}$$
So,
$$-13$$
Combining rational expressions
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-13 + a*(-1 + a)
$$a \left(a - 1\right) - 13$$
-13 + a*(-1 + a)