Mister Exam
Factor -x^2+12*x-42 squared
The solution
Factorization
[src]
/ ___\ / ___\ \x + -6 + I*\/ 6 /*\x + -6 - I*\/ 6 /
$$\left(x + \left(-6 - \sqrt{6} i\right)\right) \left(x + \left(-6 + \sqrt{6} i\right)\right)$$
(x - 6 + i*sqrt(6))*(x - 6 - i*sqrt(6))
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(- x^{2} + 12 x\right) - 42$$
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
$$a = -1$$
$$b = 12$$
$$c = -42$$
Then
$$m = -6$$
$$n = -6$$
So,
$$- \left(x - 6\right)^{2} - 6$$
$$\left(- x^{2} + 12 x\right) - 42$$
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
$$a = -1$$
$$b = 12$$
$$c = -42$$
Then
$$m = -6$$
$$n = -6$$
So,
$$- \left(x - 6\right)^{2} - 6$$
Combining rational expressions
[src]
-42 + x*(12 - x)
$$x \left(12 - x\right) - 42$$
-42 + x*(12 - x)