Solve -n+sqrt{6n+19}=2

$$- n + \sqrt { 6 n + 19 } = 2$$

Solve for n

$$n=5$$

Solution Steps

Subtract -n from both sides of the equation.

$$\sqrt{6n+19}=2-\left(-n\right)$$

Multiply -1 and -1 to get 1.

$$\sqrt{6n+19}=2+n$$

Square both sides of the equation.

$$\left(\sqrt{6n+19}\right)^{2}=\left(2+n\right)^{2}$$

Calculate \sqrt{6n+19} to the power of 2 and get 6n+19.

$$6n+19=\left(2+n\right)^{2}$$

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

$$6n+19=4+4n+n^{2}$$

Subtract 4 from both sides.

$$6n+19-4=4n+n^{2}$$

Subtract 4 from 19 to get 15.

$$6n+15=4n+n^{2}$$

Subtract 4n from both sides.

$$6n+15-4n=n^{2}$$

Combine 6n and -4n to get 2n.

$$2n+15=n^{2}$$

Subtract n^{2} from both sides.

$$2n+15-n^{2}=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-n^{2}+2n+15=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -n^{2}+an+bn+15. To find a and b, set up a system to be solved.

$$a+b=2 ab=-15=-15$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -15.

$$-1,15 -3,5$$

Calculate the sum for each pair.

$$-1+15=14 -3+5=2$$

The solution is the pair that gives sum 2.

$$a=5 b=-3$$

Rewrite -n^{2}+2n+15 as \left(-n^{2}+5n\right)+\left(-3n+15\right).

$$\left(-n^{2}+5n\right)+\left(-3n+15\right)$$

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

$$-n\left(n-5\right)-3\left(n-5\right)$$

Factor out common term n-5 by using distributive property.

$$\left(n-5\right)\left(-n-3\right)$$

To find equation solutions, solve n-5=0 and -n-3=0.

$$n=5 n=-3$$

Substitute 5 for n in the equation -n+\sqrt{6n+19}=2.

$$-5+\sqrt{6\times 5+19}=2$$

Simplify. The value n=5 satisfies the equation.

$$2=2$$

Substitute -3 for n in the equation -n+\sqrt{6n+19}=2.

$$-\left(-3\right)+\sqrt{6\left(-3\right)+19}=2$$

Simplify. The value n=-3 does not satisfy the equation.

$$4=2$$

Equation \sqrt{6n+19}=n+2 has a unique solution.

$$n=5$$