Solve 3n^2-n-574=0

$$3 { n }^{ 2 } -n-574 = 0$$

Solve for n

$$n = -\frac{41}{3} = -13\frac{2}{3} \approx -13.666666667$$

$$n=14$$

Steps Using Factoring By Grouping

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 3n^{2}+an+bn-574. To find a and b, set up a system to be solved.

$$a+b=-1 ab=3\left(-574\right)=-1722$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -1722.

$$1,-1722 2,-861 3,-574 6,-287 7,-246 14,-123 21,-82 41,-42$$

Calculate the sum for each pair.

$$1-1722=-1721 2-861=-859 3-574=-571 6-287=-281 7-246=-239 14-123=-109 21-82=-61 41-42=-1$$

The solution is the pair that gives sum -1.

$$a=-42 b=41$$

Rewrite 3n^{2}-n-574 as \left(3n^{2}-42n\right)+\left(41n-574\right).

$$\left(3n^{2}-42n\right)+\left(41n-574\right)$$

Factor out 3n in the first and 41 in the second group.

$$3n\left(n-14\right)+41\left(n-14\right)$$

Factor out common term n-14 by using distributive property.

$$\left(n-14\right)\left(3n+41\right)$$

To find equation solutions, solve n-14=0 and 3n+41=0.

$$n=14 n=-\frac{41}{3}$$

Steps Using the Quadratic Formula

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.

$$3n^{2}-n-574=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, -1 for b, and -574 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-\left(-1\right)±\sqrt{1-4\times 3\left(-574\right)}}{2\times 3}$$

Multiply -4 times 3.

$$n=\frac{-\left(-1\right)±\sqrt{1-12\left(-574\right)}}{2\times 3}$$

Multiply -12 times -574.

$$n=\frac{-\left(-1\right)±\sqrt{1+6888}}{2\times 3}$$

Add 1 to 6888.

$$n=\frac{-\left(-1\right)±\sqrt{6889}}{2\times 3}$$

Take the square root of 6889.

$$n=\frac{-\left(-1\right)±83}{2\times 3}$$

The opposite of -1 is 1.

$$n=\frac{1±83}{2\times 3}$$

Multiply 2 times 3.

$$n=\frac{1±83}{6}$$

Now solve the equation n=\frac{1±83}{6} when ± is plus. Add 1 to 83.

$$n=\frac{84}{6}$$

Divide 84 by 6.

$$n=14$$

Now solve the equation n=\frac{1±83}{6} when ± is minus. Subtract 83 from 1.

$$n=-\frac{82}{6}$$

Reduce the fraction \frac{-82}{6} to lowest terms by extracting and canceling out 2.

$$n=-\frac{41}{3}$$

The equation is now solved.

$$n=14 n=-\frac{41}{3}$$

Steps for Completing the Square

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$3n^{2}-n-574=0$$

Add 574 to both sides of the equation.

$$3n^{2}-n-574-\left(-574\right)=-\left(-574\right)$$

Subtracting -574 from itself leaves 0.

$$3n^{2}-n=-\left(-574\right)$$

Subtract -574 from 0.

$$3n^{2}-n=574$$

Divide both sides by 3.

$$\frac{3n^{2}-n}{3}=\frac{574}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$n^{2}-\frac{1}{3}n=\frac{574}{3}$$

Divide -\frac{1}{3}, the coefficient of the x term, by 2 to get -\frac{1}{6}. Then add the square of -\frac{1}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}-\frac{1}{3}n+\left(-\frac{1}{6}\right)^{2}=\frac{574}{3}+\left(-\frac{1}{6}\right)^{2}$$

Square -\frac{1}{6} by squaring both the numerator and the denominator of the fraction.

$$n^{2}-\frac{1}{3}n+\frac{1}{36}=\frac{574}{3}+\frac{1}{36}$$

Add \frac{574}{3} to \frac{1}{36} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$n^{2}-\frac{1}{3}n+\frac{1}{36}=\frac{6889}{36}$$

Factor n^{2}-\frac{1}{3}n+\frac{1}{36}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(n-\frac{1}{6}\right)^{2}=\frac{6889}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n-\frac{1}{6}\right)^{2}}=\sqrt{\frac{6889}{36}}$$

Simplify.

$$n-\frac{1}{6}=\frac{83}{6} n-\frac{1}{6}=-\frac{83}{6}$$

Add \frac{1}{6} to both sides of the equation.

$$n=14 n=-\frac{41}{3}$$