Solve 3n^2+47n-232=5

$$3 n ^ { 2 } + 47 n - 232 = 5$$

Solve for n

$$n = \frac{\sqrt{5053} - 47}{6} \approx 4.014076135$$

$$n=\frac{-\sqrt{5053}-47}{6}\approx -19.680742802$$

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.

$$3n^{2}+47n-232=5$$

Subtract 5 from both sides of the equation.

$$3n^{2}+47n-232-5=5-5$$

Subtracting 5 from itself leaves 0.

$$3n^{2}+47n-232-5=0$$

Subtract 5 from -232.

$$3n^{2}+47n-237=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, 47 for b, and -237 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-47±\sqrt{47^{2}-4\times 3\left(-237\right)}}{2\times 3}$$

Square 47.

$$n=\frac{-47±\sqrt{2209-4\times 3\left(-237\right)}}{2\times 3}$$

Multiply -4 times 3.

$$n=\frac{-47±\sqrt{2209-12\left(-237\right)}}{2\times 3}$$

Multiply -12 times -237.

$$n=\frac{-47±\sqrt{2209+2844}}{2\times 3}$$

Add 2209 to 2844.

$$n=\frac{-47±\sqrt{5053}}{2\times 3}$$

Multiply 2 times 3.

$$n=\frac{-47±\sqrt{5053}}{6}$$

Now solve the equation n=\frac{-47±\sqrt{5053}}{6} when ± is plus. Add -47 to \sqrt{5053}.

$$n=\frac{\sqrt{5053}-47}{6}$$

Now solve the equation n=\frac{-47±\sqrt{5053}}{6} when ± is minus. Subtract \sqrt{5053} from -47.

$$n=\frac{-\sqrt{5053}-47}{6}$$

The equation is now solved.

$$n=\frac{\sqrt{5053}-47}{6} n=\frac{-\sqrt{5053}-47}{6}$$

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.

$$3n^{2}+47n-232=5$$

Add 232 to both sides of the equation.

$$3n^{2}+47n-232-\left(-232\right)=5-\left(-232\right)$$

Subtracting -232 from itself leaves 0.

$$3n^{2}+47n=5-\left(-232\right)$$

Subtract -232 from 5.

$$3n^{2}+47n=237$$

Divide both sides by 3.

$$\frac{3n^{2}+47n}{3}=\frac{237}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$n^{2}+\frac{47}{3}n=\frac{237}{3}$$

Divide 237 by 3.

$$n^{2}+\frac{47}{3}n=79$$

Divide \frac{47}{3}, the coefficient of the x term, by 2 to get \frac{47}{6}. Then add the square of \frac{47}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}+\frac{47}{3}n+\left(\frac{47}{6}\right)^{2}=79+\left(\frac{47}{6}\right)^{2}$$

Square \frac{47}{6} by squaring both the numerator and the denominator of the fraction.

$$n^{2}+\frac{47}{3}n+\frac{2209}{36}=79+\frac{2209}{36}$$

Add 79 to \frac{2209}{36}.

$$n^{2}+\frac{47}{3}n+\frac{2209}{36}=\frac{5053}{36}$$

Factor n^{2}+\frac{47}{3}n+\frac{2209}{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}.

$$\left(n+\frac{47}{6}\right)^{2}=\frac{5053}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n+\frac{47}{6}\right)^{2}}=\sqrt{\frac{5053}{36}}$$

Simplify.

$$n+\frac{47}{6}=\frac{\sqrt{5053}}{6} n+\frac{47}{6}=-\frac{\sqrt{5053}}{6}$$

Subtract \frac{47}{6} from both sides of the equation.

$$n=\frac{\sqrt{5053}-47}{6} n=\frac{-\sqrt{5053}-47}{6}$$