Mister Exam
Factor 5*x^2-(25*x^2-10*x+1)/2-x^2-(x^2-2*x+1)/2 squared
The solution
You have entered
[src]
2 2
2 25*x - 10*x + 1 2 x - 2*x + 1
5*x - ---------------- - x - ------------
2 2
$$\left(- x^{2} + \left(5 x^{2} - \frac{\left(25 x^{2} - 10 x\right) + 1}{2}\right)\right) - \frac{\left(x^{2} - 2 x\right) + 1}{2}$$
5*x^2 - (25*x^2 - 10*x + 1)/2 - x^2 - (x^2 - 2*x + 1)/2
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(- x^{2} + \left(5 x^{2} - \frac{\left(25 x^{2} - 10 x\right) + 1}{2}\right)\right) - \frac{\left(x^{2} - 2 x\right) + 1}{2}$$
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
$$a = -9$$
$$b = 6$$
$$c = -1$$
Then
$$m = - \frac{1}{3}$$
$$n = 0$$
So,
$$- 9 \left(x - \frac{1}{3}\right)^{2}$$
$$\left(- x^{2} + \left(5 x^{2} - \frac{\left(25 x^{2} - 10 x\right) + 1}{2}\right)\right) - \frac{\left(x^{2} - 2 x\right) + 1}{2}$$
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
$$a = -9$$
$$b = 6$$
$$c = -1$$
Then
$$m = - \frac{1}{3}$$
$$n = 0$$
So,
$$- 9 \left(x - \frac{1}{3}\right)^{2}$$
Rational denominator
[src]
2
-2 - 18*x + 12*x
-----------------
2
$$\frac{- 18 x^{2} + 12 x - 2}{2}$$
(-2 - 18*x^2 + 12*x)/2
Combining rational expressions
[src]
2
-2 + 8*x - x*(-2 + x) - 5*x*(-2 + 5*x)
---------------------------------------
2
$$\frac{8 x^{2} - x \left(x - 2\right) - 5 x \left(5 x - 2\right) - 2}{2}$$
(-2 + 8*x^2 - x*(-2 + x) - 5*x*(-2 + 5*x))/2