Mister Exam
Factor x^2+9*x+5 squared
The solution
Factorization
[src]
/ ____\ / ____\ | 9 \/ 61 | | 9 \/ 61 | |x + - + ------|*|x + - - ------| \ 2 2 / \ 2 2 /
$$\left(x + \left(\frac{9}{2} - \frac{\sqrt{61}}{2}\right)\right) \left(x + \left(\frac{\sqrt{61}}{2} + \frac{9}{2}\right)\right)$$
(x + 9/2 + sqrt(61)/2)*(x + 9/2 - sqrt(61)/2)
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(x^{2} + 9 x\right) + 5$$
To do this, let's use the formula
$$a x^{2} + b x + c = a \left(m + x\right)^{2} + n$$
where
$$m = \frac{b}{2 a}$$
$$n = \frac{4 a c - b^{2}}{4 a}$$
In this case
$$a = 1$$
$$b = 9$$
$$c = 5$$
Then
$$m = \frac{9}{2}$$
$$n = - \frac{61}{4}$$
So,
$$\left(x + \frac{9}{2}\right)^{2} - \frac{61}{4}$$
$$\left(x^{2} + 9 x\right) + 5$$
To do this, let's use the formula
$$a x^{2} + b x + c = a \left(m + x\right)^{2} + n$$
where
$$m = \frac{b}{2 a}$$
$$n = \frac{4 a c - b^{2}}{4 a}$$
In this case
$$a = 1$$
$$b = 9$$
$$c = 5$$
Then
$$m = \frac{9}{2}$$
$$n = - \frac{61}{4}$$
So,
$$\left(x + \frac{9}{2}\right)^{2} - \frac{61}{4}$$