Mister Exam
Factor -a^2-3*a+7 squared
The solution
Factorization
[src]
/ ____\ / ____\ | 3 \/ 37 | | 3 \/ 37 | |a + - - ------|*|a + - + ------| \ 2 2 / \ 2 2 /
$$\left(a + \left(\frac{3}{2} - \frac{\sqrt{37}}{2}\right)\right) \left(a + \left(\frac{3}{2} + \frac{\sqrt{37}}{2}\right)\right)$$
(a + 3/2 - sqrt(37)/2)*(a + 3/2 + sqrt(37)/2)
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(- a^{2} - 3 a\right) + 7$$
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
$$a = -1$$
$$b = -3$$
$$c = 7$$
Then
$$m = \frac{3}{2}$$
$$n = \frac{37}{4}$$
So,
$$9$$
$$\left(- a^{2} - 3 a\right) + 7$$
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
$$a = -1$$
$$b = -3$$
$$c = 7$$
Then
$$m = \frac{3}{2}$$
$$n = \frac{37}{4}$$
So,
$$9$$