Solve -(2+n)^2+4(2+n)=-6

$$- ( 2 + n ) ^ { 2 } + 4 ( 2 + n ) = - 6$$

Solve for n

$$n=\sqrt{10}\approx 3.16227766$$

$$n=-\sqrt{10}\approx -3.16227766$$

Steps by Finding Square Root

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2+n\right)^{2}.

$$-\left(4+4n+n^{2}\right)+4\left(2+n\right)=-6$$

To find the opposite of 4+4n+n^{2}, find the opposite of each term.

$$-4-4n-n^{2}+4\left(2+n\right)=-6$$

Use the distributive property to multiply 4 by 2+n.

$$-4-4n-n^{2}+8+4n=-6$$

Add -4 and 8 to get 4.

$$4-4n-n^{2}+4n=-6$$

Combine -4n and 4n to get 0.

$$4-n^{2}=-6$$

Subtract 4 from both sides.

$$-n^{2}=-6-4$$

Subtract 4 from -6 to get -10.

$$-n^{2}=-10$$

Divide both sides by -1.

$$n^{2}=\frac{-10}{-1}$$

Fraction \frac{-10}{-1} can be simplified to 10 by removing the negative sign from both the numerator and the denominator.

$$n^{2}=10$$

Take the square root of both sides of the equation.

$$n=\sqrt{10} n=-\sqrt{10}$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2+n\right)^{2}.

$$-\left(4+4n+n^{2}\right)+4\left(2+n\right)=-6$$

To find the opposite of 4+4n+n^{2}, find the opposite of each term.

$$-4-4n-n^{2}+4\left(2+n\right)=-6$$

Use the distributive property to multiply 4 by 2+n.

$$-4-4n-n^{2}+8+4n=-6$$

Add -4 and 8 to get 4.

$$4-4n-n^{2}+4n=-6$$

Combine -4n and 4n to get 0.

$$4-n^{2}=-6$$

Add 6 to both sides.

$$4-n^{2}+6=0$$

Add 4 and 6 to get 10.

$$10-n^{2}=0$$

Quadratic equations like this one, with an x^{2} term but no x term, can still be solved using the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}, once they are put in standard form: ax^{2}+bx+c=0.

$$-n^{2}+10=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 0 for b, and 10 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{0±\sqrt{0^{2}-4\left(-1\right)\times 10}}{2\left(-1\right)}$$

Square 0.

$$n=\frac{0±\sqrt{-4\left(-1\right)\times 10}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$n=\frac{0±\sqrt{4\times 10}}{2\left(-1\right)}$$

Multiply 4 times 10.

$$n=\frac{0±\sqrt{40}}{2\left(-1\right)}$$

Take the square root of 40.

$$n=\frac{0±2\sqrt{10}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$n=\frac{0±2\sqrt{10}}{-2}$$

Now solve the equation n=\frac{0±2\sqrt{10}}{-2} when ± is plus.

$$n=-\sqrt{10}$$

Now solve the equation n=\frac{0±2\sqrt{10}}{-2} when ± is minus.

$$n=\sqrt{10}$$

The equation is now solved.

$$n=-\sqrt{10} n=\sqrt{10}$$