Solve 3n^2-2n-512/4n+1n=0

$$\frac { 3 n ^ { 2 } - 2 n - 512 } { 4 n + 1 n } = 0$$

Solve for n

$$n = \frac{\sqrt{1537} + 1}{3} \approx 13.401530523$$

$$n=\frac{1-\sqrt{1537}}{3}\approx -12.734863856$$

Steps Using the Quadratic Formula

Variable n cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 5n.

$$3n^{2}-2n-512=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, -2 for b, and -512 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\times 3\left(-512\right)}}{2\times 3}$$

Square -2.

$$n=\frac{-\left(-2\right)±\sqrt{4-4\times 3\left(-512\right)}}{2\times 3}$$

Multiply -4 times 3.

$$n=\frac{-\left(-2\right)±\sqrt{4-12\left(-512\right)}}{2\times 3}$$

Multiply -12 times -512.

$$n=\frac{-\left(-2\right)±\sqrt{4+6144}}{2\times 3}$$

Add 4 to 6144.

$$n=\frac{-\left(-2\right)±\sqrt{6148}}{2\times 3}$$

Take the square root of 6148.

$$n=\frac{-\left(-2\right)±2\sqrt{1537}}{2\times 3}$$

The opposite of -2 is 2.

$$n=\frac{2±2\sqrt{1537}}{2\times 3}$$

Multiply 2 times 3.

$$n=\frac{2±2\sqrt{1537}}{6}$$

Now solve the equation n=\frac{2±2\sqrt{1537}}{6} when ± is plus. Add 2 to 2\sqrt{1537}.

$$n=\frac{2\sqrt{1537}+2}{6}$$

Divide 2+2\sqrt{1537} by 6.

$$n=\frac{\sqrt{1537}+1}{3}$$

Now solve the equation n=\frac{2±2\sqrt{1537}}{6} when ± is minus. Subtract 2\sqrt{1537} from 2.

$$n=\frac{2-2\sqrt{1537}}{6}$$

Divide 2-2\sqrt{1537} by 6.

$$n=\frac{1-\sqrt{1537}}{3}$$

The equation is now solved.

$$n=\frac{\sqrt{1537}+1}{3} n=\frac{1-\sqrt{1537}}{3}$$

Steps for Completing the Square

Variable n cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 5n.

$$3n^{2}-2n-512=0$$

Add 512 to both sides. Anything plus zero gives itself.

$$3n^{2}-2n=512$$

Divide both sides by 3.

$$\frac{3n^{2}-2n}{3}=\frac{512}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$n^{2}-\frac{2}{3}n=\frac{512}{3}$$

Divide -\frac{2}{3}, the coefficient of the x term, by 2 to get -\frac{1}{3}. Then add the square of -\frac{1}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}-\frac{2}{3}n+\left(-\frac{1}{3}\right)^{2}=\frac{512}{3}+\left(-\frac{1}{3}\right)^{2}$$

Square -\frac{1}{3} by squaring both the numerator and the denominator of the fraction.

$$n^{2}-\frac{2}{3}n+\frac{1}{9}=\frac{512}{3}+\frac{1}{9}$$

Add \frac{512}{3} to \frac{1}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$n^{2}-\frac{2}{3}n+\frac{1}{9}=\frac{1537}{9}$$

Factor n^{2}-\frac{2}{3}n+\frac{1}{9}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(n-\frac{1}{3}\right)^{2}=\frac{1537}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n-\frac{1}{3}\right)^{2}}=\sqrt{\frac{1537}{9}}$$

Simplify.

$$n-\frac{1}{3}=\frac{\sqrt{1537}}{3} n-\frac{1}{3}=-\frac{\sqrt{1537}}{3}$$

Add \frac{1}{3} to both sides of the equation.

$$n=\frac{\sqrt{1537}+1}{3} n=\frac{1-\sqrt{1537}}{3}$$