Solve -3+sqrt{4n+1}=sqrt{4-2n}

$$- 3 + \sqrt { 4 n + 1 } = \sqrt { 4 - 2 n }$$

Solve for n

$$n=2$$

Solution Steps

Square both sides of the equation.

$$\left(-3+\sqrt{4n+1}\right)^{2}=\left(\sqrt{4-2n}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(-3+\sqrt{4n+1}\right)^{2}.

$$9-6\sqrt{4n+1}+\left(\sqrt{4n+1}\right)^{2}=\left(\sqrt{4-2n}\right)^{2}$$

Calculate \sqrt{4n+1} to the power of 2 and get 4n+1.

$$9-6\sqrt{4n+1}+4n+1=\left(\sqrt{4-2n}\right)^{2}$$

Add 9 and 1 to get 10.

$$10-6\sqrt{4n+1}+4n=\left(\sqrt{4-2n}\right)^{2}$$

Calculate \sqrt{4-2n} to the power of 2 and get 4-2n.

$$10-6\sqrt{4n+1}+4n=4-2n$$

Subtract 10+4n from both sides of the equation.

$$-6\sqrt{4n+1}=4-2n-\left(10+4n\right)$$

To find the opposite of 10+4n, find the opposite of each term.

$$-6\sqrt{4n+1}=4-2n-10-4n$$

Subtract 10 from 4 to get -6.

$$-6\sqrt{4n+1}=-6-2n-4n$$

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

$$-6\sqrt{4n+1}=-6-6n$$

Square both sides of the equation.

$$\left(-6\sqrt{4n+1}\right)^{2}=\left(-6-6n\right)^{2}$$

Expand \left(-6\sqrt{4n+1}\right)^{2}.

$$\left(-6\right)^{2}\left(\sqrt{4n+1}\right)^{2}=\left(-6-6n\right)^{2}$$

Calculate -6 to the power of 2 and get 36.

$$36\left(\sqrt{4n+1}\right)^{2}=\left(-6-6n\right)^{2}$$

Calculate \sqrt{4n+1} to the power of 2 and get 4n+1.

$$36\left(4n+1\right)=\left(-6-6n\right)^{2}$$

Use the distributive property to multiply 36 by 4n+1.

$$144n+36=\left(-6-6n\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(-6-6n\right)^{2}.

$$144n+36=36+72n+36n^{2}$$

Subtract 36 from both sides.

$$144n+36-36=72n+36n^{2}$$

Subtract 36 from 36 to get 0.

$$144n=72n+36n^{2}$$

Subtract 72n from both sides.

$$144n-72n=36n^{2}$$

Combine 144n and -72n to get 72n.

$$72n=36n^{2}$$

Subtract 36n^{2} from both sides.

$$72n-36n^{2}=0$$

Factor out n.

$$n\left(72-36n\right)=0$$

To find equation solutions, solve n=0 and 72-36n=0.

$$n=0 n=2$$

Substitute 0 for n in the equation -3+\sqrt{4n+1}=\sqrt{4-2n}.

$$-3+\sqrt{4\times 0+1}=\sqrt{4-2\times 0}$$

Simplify. The value n=0 does not satisfy the equation because the left and the right hand side have opposite signs.

$$-2=2$$

Substitute 2 for n in the equation -3+\sqrt{4n+1}=\sqrt{4-2n}.

$$-3+\sqrt{4\times 2+1}=\sqrt{4-2\times 2}$$

Simplify. The value n=2 satisfies the equation.

$$0=0$$

Equation \sqrt{4n+1}-3=\sqrt{4-2n} has a unique solution.

$$n=2$$