Mister Exam
Factor 3*x^2+5*x-3 squared
The solution
Factorization
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/ ____\ / ____\ | 5 \/ 61 | | 5 \/ 61 | |x + - - ------|*|x + - + ------| \ 6 6 / \ 6 6 /
$$\left(x + \left(\frac{5}{6} - \frac{\sqrt{61}}{6}\right)\right) \left(x + \left(\frac{5}{6} + \frac{\sqrt{61}}{6}\right)\right)$$
(x + 5/6 - sqrt(61)/6)*(x + 5/6 + sqrt(61)/6)
The perfect square
Let's highlight the perfect square of the square three-member
$$\left(3 x^{2} + 5 x\right) - 3$$
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
$$a = 3$$
$$b = 5$$
$$c = -3$$
Then
$$m = \frac{5}{6}$$
$$n = - \frac{61}{12}$$
So,
$$3 \left(x + \frac{5}{6}\right)^{2} - \frac{61}{12}$$
$$\left(3 x^{2} + 5 x\right) - 3$$
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
$$a = 3$$
$$b = 5$$
$$c = -3$$
Then
$$m = \frac{5}{6}$$
$$n = - \frac{61}{12}$$
So,
$$3 \left(x + \frac{5}{6}\right)^{2} - \frac{61}{12}$$
Combining rational expressions
[src]
-3 + x*(5 + 3*x)
$$x \left(3 x + 5\right) - 3$$
-3 + x*(5 + 3*x)