Mister Exam
Factor polynomial x^2-15*x+63
The solution
Factorization
[src]
/ ___\ / ___\ | 15 3*I*\/ 3 | | 15 3*I*\/ 3 | |x + - -- + ---------|*|x + - -- - ---------| \ 2 2 / \ 2 2 /
$$\left(x + \left(- \frac{15}{2} - \frac{3 \sqrt{3} i}{2}\right)\right) \left(x + \left(- \frac{15}{2} + \frac{3 \sqrt{3} i}{2}\right)\right)$$
(x - 15/2 + 3*i*sqrt(3)/2)*(x - 15/2 - 3*i*sqrt(3)/2)
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(x^{2} - 15 x\right) + 63$$
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
$$a = 1$$
$$b = -15$$
$$c = 63$$
Then
$$m = - \frac{15}{2}$$
$$n = \frac{27}{4}$$
So,
$$\left(x - \frac{15}{2}\right)^{2} + \frac{27}{4}$$
$$\left(x^{2} - 15 x\right) + 63$$
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
$$a = 1$$
$$b = -15$$
$$c = 63$$
Then
$$m = - \frac{15}{2}$$
$$n = \frac{27}{4}$$
So,
$$\left(x - \frac{15}{2}\right)^{2} + \frac{27}{4}$$
Combining rational expressions
[src]
63 + x*(-15 + x)
$$x \left(x - 15\right) + 63$$
63 + x*(-15 + x)