Solve 3x^2+5x+2

$$3 { x }^{ 2 } +5x+2$$

Factor

$$\left(x+1\right)\left(3x+2\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 3x^{2}+ax+bx+2. To find a and b, set up a system to be solved.

$$a+b=5 ab=3\times 2=6$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 6.

$$1,6 2,3$$

Calculate the sum for each pair.

$$1+6=7 2+3=5$$

The solution is the pair that gives sum 5.

$$a=2 b=3$$

Rewrite 3x^{2}+5x+2 as \left(3x^{2}+2x\right)+\left(3x+2\right).

$$\left(3x^{2}+2x\right)+\left(3x+2\right)$$

Factor out x in 3x^{2}+2x.

$$x\left(3x+2\right)+3x+2$$

Factor out common term 3x+2 by using distributive property.

$$\left(3x+2\right)\left(x+1\right)$$

Steps Using the Quadratic Formula

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$3x^{2}+5x+2=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$x=\frac{-5±\sqrt{5^{2}-4\times 3\times 2}}{2\times 3}$$

Square 5.

$$x=\frac{-5±\sqrt{25-4\times 3\times 2}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-5±\sqrt{25-12\times 2}}{2\times 3}$$

Multiply -12 times 2.

$$x=\frac{-5±\sqrt{25-24}}{2\times 3}$$

Add 25 to -24.

$$x=\frac{-5±\sqrt{1}}{2\times 3}$$

Take the square root of 1.

$$x=\frac{-5±1}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{-5±1}{6}$$

Now solve the equation x=\frac{-5±1}{6} when ± is plus. Add -5 to 1.

$$x=-\frac{4}{6}$$

Reduce the fraction \frac{-4}{6} to lowest terms by extracting and canceling out 2.

$$x=-\frac{2}{3}$$

Now solve the equation x=\frac{-5±1}{6} when ± is minus. Subtract 1 from -5.

$$x=-\frac{6}{6}$$

Divide -6 by 6.

$$x=-1$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\frac{2}{3} for x_{1} and -1 for x_{2}.

$$3x^{2}+5x+2=3\left(x-\left(-\frac{2}{3}\right)\right)\left(x-\left(-1\right)\right)$$

Simplify all the expressions of the form p-\left(-q\right) to p+q.

$$3x^{2}+5x+2=3\left(x+\frac{2}{3}\right)\left(x+1\right)$$

Add \frac{2}{3} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$3x^{2}+5x+2=3\times \frac{3x+2}{3}\left(x+1\right)$$

Cancel out 3, the greatest common factor in 3 and 3.

$$3x^{2}+5x+2=\left(3x+2\right)\left(x+1\right)$$

Evaluate

$$\left(x+1\right)\left(3x+2\right)$$