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

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

Solve for n

$$n=5-2\sqrt{10}\approx -1.32455532$$

Solution Steps

Subtract n from both sides of the equation.

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

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.

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

Add 4n to both sides.

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

Combine 6n and 4n to get 10n.

$$10n+15=n^{2}$$

Subtract n^{2} from both sides.

$$10n+15-n^{2}=0$$

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.

$$-n^{2}+10n+15=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 10 for b, and 15 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-10±\sqrt{10^{2}-4\left(-1\right)\times 15}}{2\left(-1\right)}$$

Square 10.

$$n=\frac{-10±\sqrt{100-4\left(-1\right)\times 15}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$n=\frac{-10±\sqrt{100+4\times 15}}{2\left(-1\right)}$$

Multiply 4 times 15.

$$n=\frac{-10±\sqrt{100+60}}{2\left(-1\right)}$$

Add 100 to 60.

$$n=\frac{-10±\sqrt{160}}{2\left(-1\right)}$$

Take the square root of 160.

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

Multiply 2 times -1.

$$n=\frac{-10±4\sqrt{10}}{-2}$$

Now solve the equation n=\frac{-10±4\sqrt{10}}{-2} when ± is plus. Add -10 to 4\sqrt{10}.

$$n=\frac{4\sqrt{10}-10}{-2}$$

Divide -10+4\sqrt{10} by -2.

$$n=5-2\sqrt{10}$$

Now solve the equation n=\frac{-10±4\sqrt{10}}{-2} when ± is minus. Subtract 4\sqrt{10} from -10.

$$n=\frac{-4\sqrt{10}-10}{-2}$$

Divide -10-4\sqrt{10} by -2.

$$n=2\sqrt{10}+5$$

The equation is now solved.

$$n=5-2\sqrt{10} n=2\sqrt{10}+5$$

Substitute 5-2\sqrt{10} for n in the equation n+\sqrt{6n+19}=2.

$$5-2\sqrt{10}+\sqrt{6\left(5-2\sqrt{10}\right)+19}=2$$

Simplify. The value n=5-2\sqrt{10} satisfies the equation.

$$2=2$$

Substitute 2\sqrt{10}+5 for n in the equation n+\sqrt{6n+19}=2.

$$2\sqrt{10}+5+\sqrt{6\left(2\sqrt{10}+5\right)+19}=2$$

Simplify. The value n=2\sqrt{10}+5 does not satisfy the equation.

$$4\times 10^{\frac{1}{2}}+8=2$$

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

$$n=5-2\sqrt{10}$$