Mister Exam

# Factor 3*x^2+5*x+3 squared

An expression to simplify:

### The solution

You have entered [src]
   2
3*x  + 5*x + 3
$$\left(3 x^{2} + 5 x\right) + 3$$
3*x^2 + 5*x + 3
Factorization [src]
/            ____\ /            ____\
|    5   I*\/ 11 | |    5   I*\/ 11 |
|x + - + --------|*|x + - - --------|
\    6      6    / \    6      6    /
$$\left(x + \left(\frac{5}{6} - \frac{\sqrt{11} i}{6}\right)\right) \left(x + \left(\frac{5}{6} + \frac{\sqrt{11} i}{6}\right)\right)$$
(x + 5/6 + i*sqrt(11)/6)*(x + 5/6 - i*sqrt(11)/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
$$m = \frac{5}{6}$$
$$n = \frac{11}{12}$$
So,
$$3 \left(x + \frac{5}{6}\right)^{2} + \frac{11}{12}$$
General simplification [src]
       2
3 + 3*x  + 5*x
$$3 x^{2} + 5 x + 3$$
3 + 3*x^2 + 5*x
Assemble expression [src]
       2
3 + 3*x  + 5*x
$$3 x^{2} + 5 x + 3$$
3 + 3*x^2 + 5*x
Rational denominator [src]
       2
3 + 3*x  + 5*x
$$3 x^{2} + 5 x + 3$$
3 + 3*x^2 + 5*x
3.0 + 3.0*x^2 + 5.0*x
3.0 + 3.0*x^2 + 5.0*x
Combining rational expressions [src]
3 + x*(5 + 3*x)
$$x \left(3 x + 5\right) + 3$$
3 + x*(5 + 3*x)
Powers [src]
       2
3 + 3*x  + 5*x
$$3 x^{2} + 5 x + 3$$
3 + 3*x^2 + 5*x
Trigonometric part [src]
       2
3 + 3*x  + 5*x
$$3 x^{2} + 5 x + 3$$
3 + 3*x^2 + 5*x
Common denominator [src]
       2
3 + 3*x  + 5*x
$$3 x^{2} + 5 x + 3$$
3 + 3*x^2 + 5*x
Combinatorics [src]
       2
3 + 3*x  + 5*x
$$3 x^{2} + 5 x + 3$$
3 + 3*x^2 + 5*x