Solve 3x^2+7x+2=

$$3 { x }^{ 2 } +7x+2=$$

Factor

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

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.

$$a+b=7 ab=3\times 2=6$$

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 7.

$$a=1 b=6$$

Rewrite 3x^{2}+7x+2 as \left(3x^{2}+x\right)+\left(6x+2\right).

$$\left(3x^{2}+x\right)+\left(6x+2\right)$$

Factor out x in the first and 2 in the second group.

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

Factor out common term 3x+1 by using distributive property.

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

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.

$$3x^{2}+7x+2=0$$

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.

$$x=\frac{-7±\sqrt{7^{2}-4\times 3\times 2}}{2\times 3}$$

Square 7.

$$x=\frac{-7±\sqrt{49-4\times 3\times 2}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-7±\sqrt{49-12\times 2}}{2\times 3}$$

Multiply -12 times 2.

$$x=\frac{-7±\sqrt{49-24}}{2\times 3}$$

Add 49 to -24.

$$x=\frac{-7±\sqrt{25}}{2\times 3}$$

Take the square root of 25.

$$x=\frac{-7±5}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{-7±5}{6}$$

Now solve the equation x=\frac{-7±5}{6} when ± is plus. Add -7 to 5.

$$x=-\frac{2}{6}$$

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

$$x=-\frac{1}{3}$$

Now solve the equation x=\frac{-7±5}{6} when ± is minus. Subtract 5 from -7.

$$x=-\frac{12}{6}$$

Divide -12 by 6.

$$x=-2$$

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

$$3x^{2}+7x+2=3\left(x-\left(-\frac{1}{3}\right)\right)\left(x-\left(-2\right)\right)$$

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

$$3x^{2}+7x+2=3\left(x+\frac{1}{3}\right)\left(x+2\right)$$

Add \frac{1}{3} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$3x^{2}+7x+2=3\times \frac{3x+1}{3}\left(x+2\right)$$

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

$$3x^{2}+7x+2=\left(3x+1\right)\left(x+2\right)$$

Evaluate

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