Solve frac{sqrt{12a-3a^2}}{2a}+3/2=3

$$\frac{ \sqrt{ 12a-3 { a }^{ 2 } } }{ 2a } + \frac{ 3 }{ 2 } = 3$$

Solve for a

$$a=1$$

Solution Steps

Subtract \frac{3}{2} from both sides of the equation.

$$\frac{\sqrt{12a-3a^{2}}}{2a}=\frac{3}{2}$$

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 2a,2.

$$\sqrt{12a-3a^{2}}=3a$$

Square both sides of the equation.

$$\left(\sqrt{12a-3a^{2}}\right)^{2}=\left(3a\right)^{2}$$

Calculate \sqrt{12a-3a^{2}} to the power of 2 and get 12a-3a^{2}.

$$12a-3a^{2}=\left(3a\right)^{2}$$

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

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

Calculate 3 to the power of 2 and get 9.

$$12a-3a^{2}=9a^{2}$$

Subtract 9a^{2} from both sides.

$$12a-3a^{2}-9a^{2}=0$$

Combine -3a^{2} and -9a^{2} to get -12a^{2}.

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

Factor out a.

$$a\left(12-12a\right)=0$$

To find equation solutions, solve a=0 and 12-12a=0.

$$a=0 a=1$$

Substitute 0 for a in the equation \frac{\sqrt{12a-3a^{2}}}{2a}+\frac{3}{2}=3. The expression is undefined.

$$\frac{\sqrt{12\times 0-3\times 0^{2}}}{2\times 0}+\frac{3}{2}=3$$

Substitute 1 for a in the equation \frac{\sqrt{12a-3a^{2}}}{2a}+\frac{3}{2}=3.

$$\frac{\sqrt{12\times 1-3\times 1^{2}}}{2\times 1}+\frac{3}{2}=3$$

Simplify. The value a=1 satisfies the equation.

$$3=3$$

Equation \sqrt{12a-3a^{2}}=3a has a unique solution.

$$a=1$$