Mister Exam
Factor 2*x^2-3*x-1 squared
The solution
Factorization
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/ ____\ / ____\ | 3 \/ 17 | | 3 \/ 17 | |x + - - + ------|*|x + - - - ------| \ 4 4 / \ 4 4 /
$$\left(x + \left(- \frac{3}{4} + \frac{\sqrt{17}}{4}\right)\right) \left(x + \left(- \frac{\sqrt{17}}{4} - \frac{3}{4}\right)\right)$$
(x - 3/4 + sqrt(17)/4)*(x - 3/4 - sqrt(17)/4)
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(2 x^{2} - 3 x\right) - 1$$
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
$$a = 2$$
$$b = -3$$
$$c = -1$$
Then
$$m = - \frac{3}{4}$$
$$n = - \frac{17}{8}$$
So,
$$2 \left(x - \frac{3}{4}\right)^{2} - \frac{17}{8}$$
$$\left(2 x^{2} - 3 x\right) - 1$$
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
$$a = 2$$
$$b = -3$$
$$c = -1$$
Then
$$m = - \frac{3}{4}$$
$$n = - \frac{17}{8}$$
So,
$$2 \left(x - \frac{3}{4}\right)^{2} - \frac{17}{8}$$
Combining rational expressions
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-1 + x*(-3 + 2*x)
$$x \left(2 x - 3\right) - 1$$
-1 + x*(-3 + 2*x)