Solve 1/2=frac{sqrt{a^2-3}}{a}

$$\frac { 1 } { 2 } = \frac { \sqrt { a ^ { 2 } - 3 } } { a }$$

Solve for a

$$a=2$$

Solution Steps

Variable a cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 2a, the least common multiple of 2,a.

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

Subtract 2\sqrt{a^{2}-3} from both sides.

$$a-2\sqrt{a^{2}-3}=0$$

Subtract a from both sides of the equation.

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

Square both sides of the equation.

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

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

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

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

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

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

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

Use the distributive property to multiply 4 by a^{2}-3.

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

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

$$4a^{2}-12=\left(-1\right)^{2}a^{2}$$

Calculate -1 to the power of 2 and get 1.

$$4a^{2}-12=1a^{2}$$

Subtract 1a^{2} from both sides.

$$4a^{2}-12-a^{2}=0$$

Combine 4a^{2} and -a^{2} to get 3a^{2}.

$$3a^{2}-12=0$$

Divide both sides by 3.

$$a^{2}-4=0$$

Consider a^{2}-4. Rewrite a^{2}-4 as a^{2}-2^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(a-2\right)\left(a+2\right)=0$$

To find equation solutions, solve a-2=0 and a+2=0.

$$a=2 a=-2$$

Substitute 2 for a in the equation \frac{1}{2}=\frac{\sqrt{a^{2}-3}}{a}.

$$\frac{1}{2}=\frac{\sqrt{2^{2}-3}}{2}$$

Simplify. The value a=2 satisfies the equation.

$$\frac{1}{2}=\frac{1}{2}$$

Substitute -2 for a in the equation \frac{1}{2}=\frac{\sqrt{a^{2}-3}}{a}.

$$\frac{1}{2}=\frac{\sqrt{\left(-2\right)^{2}-3}}{-2}$$

Simplify. The value a=-2 does not satisfy the equation because the left and the right hand side have opposite signs.

$$\frac{1}{2}=-\frac{1}{2}$$

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

$$a=2$$