Solve 3n^2+4n

$$3 n ^ { 2 } + 4 n$$

Factor

$$n\left(3n+4\right)$$

Solution Steps

Factor out n.

$$n\left(3n+4\right)$$

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.

$$3n^{2}+4n=0$$

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.

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

Take the square root of 4^{2}.

$$n=\frac{-4±4}{2\times 3}$$

Multiply 2 times 3.

$$n=\frac{-4±4}{6}$$

Now solve the equation n=\frac{-4±4}{6} when ± is plus. Add -4 to 4.

$$n=\frac{0}{6}$$

Divide 0 by 6.

$$n=0$$

Now solve the equation n=\frac{-4±4}{6} when ± is minus. Subtract 4 from -4.

$$n=-\frac{8}{6}$$

Reduce the fraction \frac{-8}{6} to lowest terms by extracting and canceling out 2.

$$n=-\frac{4}{3}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 0 for x_{1} and -\frac{4}{3} for x_{2}.

$$3n^{2}+4n=3n\left(n-\left(-\frac{4}{3}\right)\right)$$

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

$$3n^{2}+4n=3n\left(n+\frac{4}{3}\right)$$

Add \frac{4}{3} to n by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$3n^{2}+4n=3n\times \frac{3n+4}{3}$$

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

$$3n^{2}+4n=n\left(3n+4\right)$$

Evaluate

$$n\left(3n+4\right)$$