Solve 1290=4n+2n^2

$$1290 = 4 n + 2 n ^ { 2 }$$

Solve for n

$$n=\sqrt{646}-1\approx 24.416530054$$

$$n=-\sqrt{646}-1\approx -26.416530054$$

Steps Using the Quadratic Formula

Swap sides so that all variable terms are on the left hand side.

$$4n+2n^{2}=1290$$

Subtract 1290 from both sides.

$$4n+2n^{2}-1290=0$$

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.

$$2n^{2}+4n-1290=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 4 for b, and -1290 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-4±\sqrt{4^{2}-4\times 2\left(-1290\right)}}{2\times 2}$$

Square 4.

$$n=\frac{-4±\sqrt{16-4\times 2\left(-1290\right)}}{2\times 2}$$

Multiply -4 times 2.

$$n=\frac{-4±\sqrt{16-8\left(-1290\right)}}{2\times 2}$$

Multiply -8 times -1290.

$$n=\frac{-4±\sqrt{16+10320}}{2\times 2}$$

Add 16 to 10320.

$$n=\frac{-4±\sqrt{10336}}{2\times 2}$$

Take the square root of 10336.

$$n=\frac{-4±4\sqrt{646}}{2\times 2}$$

Multiply 2 times 2.

$$n=\frac{-4±4\sqrt{646}}{4}$$

Now solve the equation n=\frac{-4±4\sqrt{646}}{4} when ± is plus. Add -4 to 4\sqrt{646}.

$$n=\frac{4\sqrt{646}-4}{4}$$

Divide -4+4\sqrt{646} by 4.

$$n=\sqrt{646}-1$$

Now solve the equation n=\frac{-4±4\sqrt{646}}{4} when ± is minus. Subtract 4\sqrt{646} from -4.

$$n=\frac{-4\sqrt{646}-4}{4}$$

Divide -4-4\sqrt{646} by 4.

$$n=-\sqrt{646}-1$$

The equation is now solved.

$$n=\sqrt{646}-1 n=-\sqrt{646}-1$$

Steps for Completing the Square

Swap sides so that all variable terms are on the left hand side.

$$4n+2n^{2}=1290$$

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.

$$2n^{2}+4n=1290$$

Divide both sides by 2.

$$\frac{2n^{2}+4n}{2}=\frac{1290}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$n^{2}+\frac{4}{2}n=\frac{1290}{2}$$

Divide 4 by 2.

$$n^{2}+2n=\frac{1290}{2}$$

Divide 1290 by 2.

$$n^{2}+2n=645$$

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

$$n^{2}+2n+1^{2}=645+1^{2}$$

Square 1.

$$n^{2}+2n+1=645+1$$

Add 645 to 1.

$$n^{2}+2n+1=646$$

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

$$\left(n+1\right)^{2}=646$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n+1\right)^{2}}=\sqrt{646}$$

Simplify.

$$n+1=\sqrt{646} n+1=-\sqrt{646}$$

Subtract 1 from both sides of the equation.

$$n=\sqrt{646}-1 n=-\sqrt{646}-1$$