Solve sqrt{3a}-sqrt{2a}=sqrt{a}

$$\sqrt { 3 a } - \sqrt { 2 a } = \sqrt { a }$$

Solve for a

$$a=0$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{3a}-\sqrt{2a}\right)^{2}=\left(\sqrt{a}\right)^{2}$$

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

$$\left(\sqrt{3a}\right)^{2}-2\sqrt{3a}\sqrt{2a}+\left(\sqrt{2a}\right)^{2}=\left(\sqrt{a}\right)^{2}$$

Calculate \sqrt{3a} to the power of 2 and get 3a.

$$3a-2\sqrt{3a}\sqrt{2a}+\left(\sqrt{2a}\right)^{2}=\left(\sqrt{a}\right)^{2}$$

Calculate \sqrt{2a} to the power of 2 and get 2a.

$$3a-2\sqrt{3a}\sqrt{2a}+2a=\left(\sqrt{a}\right)^{2}$$

Combine 3a and 2a to get 5a.

$$5a-2\sqrt{3a}\sqrt{2a}=\left(\sqrt{a}\right)^{2}$$

Calculate \sqrt{a} to the power of 2 and get a.

$$5a-2\sqrt{3a}\sqrt{2a}=a$$

Subtract 5a from both sides of the equation.

$$-2\sqrt{3a}\sqrt{2a}=a-5a$$

Combine a and -5a to get -4a.

$$-2\sqrt{3a}\sqrt{2a}=-4a$$

Square both sides of the equation.

$$\left(-2\sqrt{3a}\sqrt{2a}\right)^{2}=\left(-4a\right)^{2}$$

Expand \left(-2\sqrt{3a}\sqrt{2a}\right)^{2}.

$$\left(-2\right)^{2}\left(\sqrt{3a}\right)^{2}\left(\sqrt{2a}\right)^{2}=\left(-4a\right)^{2}$$

Calculate -2 to the power of 2 and get 4.

$$4\left(\sqrt{3a}\right)^{2}\left(\sqrt{2a}\right)^{2}=\left(-4a\right)^{2}$$

Calculate \sqrt{3a} to the power of 2 and get 3a.

$$4\times 3a\left(\sqrt{2a}\right)^{2}=\left(-4a\right)^{2}$$

Multiply 4 and 3 to get 12.

$$12a\left(\sqrt{2a}\right)^{2}=\left(-4a\right)^{2}$$

Calculate \sqrt{2a} to the power of 2 and get 2a.

$$12a\times 2a=\left(-4a\right)^{2}$$

Multiply 12 and 2 to get 24.

$$24aa=\left(-4a\right)^{2}$$

Multiply a and a to get a^{2}.

$$24a^{2}=\left(-4a\right)^{2}$$

Expand \left(-4a\right)^{2}.

$$24a^{2}=\left(-4\right)^{2}a^{2}$$

Calculate -4 to the power of 2 and get 16.

$$24a^{2}=16a^{2}$$

Subtract 16a^{2} from both sides.

$$24a^{2}-16a^{2}=0$$

Combine 24a^{2} and -16a^{2} to get 8a^{2}.

$$8a^{2}=0$$

Divide both sides by 8. Zero divided by any non-zero number gives zero.

$$a^{2}=0$$

Take the square root of both sides of the equation.

$$a=0 a=0$$

The equation is now solved. Solutions are the same.

$$a=0$$

Substitute 0 for a in the equation \sqrt{3a}-\sqrt{2a}=\sqrt{a}.

$$\sqrt{3\times 0}-\sqrt{2\times 0}=\sqrt{0}$$

Simplify. The value a=0 satisfies the equation.

$$0=0$$

Equation \sqrt{3a}-\sqrt{2a}=\sqrt{a} has a unique solution.

$$a=0$$