Solve sqrt{3/4}a^2=3*1/2*sqrt{400-a^2/2}

$$\sqrt { \frac { 3 } { 4 } } a ^ { 2 } = 3 \times \frac { 1 } { 2 } \times \sqrt { 400 - \frac { a ^ { 2 } } { 2 } }$$

Solve for a

$$a = \frac{\sqrt{\sqrt{19209} - 3}}{2} \approx 5.822296299$$

$$a = -\frac{\sqrt{\sqrt{19209} - 3}}{2} \approx -5.822296299$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{\frac{3}{4}}a^{2}\right)^{2}=\left(3\times \frac{1}{2}\sqrt{400-\frac{a^{2}}{2}}\right)^{2}$$

Rewrite the square root of the division \sqrt{\frac{3}{4}} as the division of square roots \frac{\sqrt{3}}{\sqrt{4}}.

$$\left(\frac{\sqrt{3}}{\sqrt{4}}a^{2}\right)^{2}=\left(3\times \frac{1}{2}\sqrt{400-\frac{a^{2}}{2}}\right)^{2}$$

Calculate the square root of 4 and get 2.

$$\left(\frac{\sqrt{3}}{2}a^{2}\right)^{2}=\left(3\times \frac{1}{2}\sqrt{400-\frac{a^{2}}{2}}\right)^{2}$$

Express \frac{\sqrt{3}}{2}a^{2} as a single fraction.

$$\left(\frac{\sqrt{3}a^{2}}{2}\right)^{2}=\left(3\times \frac{1}{2}\sqrt{400-\frac{a^{2}}{2}}\right)^{2}$$

To raise \frac{\sqrt{3}a^{2}}{2} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(\sqrt{3}a^{2}\right)^{2}}{2^{2}}=\left(3\times \frac{1}{2}\sqrt{400-\frac{a^{2}}{2}}\right)^{2}$$

Multiply 3 and \frac{1}{2} to get \frac{3}{2}.

$$\frac{\left(\sqrt{3}a^{2}\right)^{2}}{2^{2}}=\left(\frac{3}{2}\sqrt{400-\frac{a^{2}}{2}}\right)^{2}$$

Expand \left(\frac{3}{2}\sqrt{400-\frac{a^{2}}{2}}\right)^{2}.

$$\frac{\left(\sqrt{3}a^{2}\right)^{2}}{2^{2}}=\left(\frac{3}{2}\right)^{2}\left(\sqrt{400-\frac{a^{2}}{2}}\right)^{2}$$

Calculate \frac{3}{2} to the power of 2 and get \frac{9}{4}.

$$\frac{\left(\sqrt{3}a^{2}\right)^{2}}{2^{2}}=\frac{9}{4}\left(\sqrt{400-\frac{a^{2}}{2}}\right)^{2}$$

Calculate \sqrt{400-\frac{a^{2}}{2}} to the power of 2 and get 400-\frac{a^{2}}{2}.

$$\frac{\left(\sqrt{3}a^{2}\right)^{2}}{2^{2}}=\frac{9}{4}\left(400-\frac{a^{2}}{2}\right)$$

Use the distributive property to multiply \frac{9}{4} by 400-\frac{a^{2}}{2}.

$$\frac{\left(\sqrt{3}a^{2}\right)^{2}}{2^{2}}=900+\frac{9}{4}\left(-\frac{a^{2}}{2}\right)$$

Multiply \frac{9}{4} times -\frac{a^{2}}{2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(\sqrt{3}a^{2}\right)^{2}}{2^{2}}=900+\frac{-9a^{2}}{4\times 2}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 900 times \frac{4\times 2}{4\times 2}.

$$\frac{\left(\sqrt{3}a^{2}\right)^{2}}{2^{2}}=\frac{900\times 4\times 2}{4\times 2}+\frac{-9a^{2}}{4\times 2}$$

Since \frac{900\times 4\times 2}{4\times 2} and \frac{-9a^{2}}{4\times 2} have the same denominator, add them by adding their numerators.

$$\frac{\left(\sqrt{3}a^{2}\right)^{2}}{2^{2}}=\frac{900\times 4\times 2-9a^{2}}{4\times 2}$$

Do the multiplications in 900\times 4\times 2-9a^{2}.

$$\frac{\left(\sqrt{3}a^{2}\right)^{2}}{2^{2}}=\frac{7200-9a^{2}}{4\times 2}$$

Multiply 4 and 2 to get 8.

$$\frac{\left(\sqrt{3}a^{2}\right)^{2}}{2^{2}}=\frac{7200-9a^{2}}{8}$$

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

$$\frac{\left(\sqrt{3}\right)^{2}\left(a^{2}\right)^{2}}{2^{2}}=\frac{7200-9a^{2}}{8}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$\frac{\left(\sqrt{3}\right)^{2}a^{4}}{2^{2}}=\frac{7200-9a^{2}}{8}$$

The square of \sqrt{3} is 3.

$$\frac{3a^{4}}{2^{2}}=\frac{7200-9a^{2}}{8}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{3a^{4}}{4}=\frac{7200-9a^{2}}{8}$$

Divide each term of 7200-9a^{2} by 8 to get 900-\frac{9}{8}a^{2}.

$$\frac{3a^{4}}{4}=900-\frac{9}{8}a^{2}$$

Multiply both sides of the equation by 8, the least common multiple of 4,8.

$$2\times 3a^{4}=7200-9a^{2}$$

Reorder the terms.

$$2\times 3a^{4}=-9a^{2}+7200$$

Multiply 2 and 3 to get 6.

$$6a^{4}=-9a^{2}+7200$$

Add 9a^{2} to both sides.

$$6a^{4}+9a^{2}=7200$$

Subtract 7200 from both sides.

$$6a^{4}+9a^{2}-7200=0$$

Substitute t for a^{2}.

$$6t^{2}+9t-7200=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Substitute 6 for a, 9 for b, and -7200 for c in the quadratic formula.

$$t=\frac{-9±\sqrt{9^{2}-4\times 6\left(-7200\right)}}{2\times 6}$$

Do the calculations.

$$t=\frac{-9±3\sqrt{19209}}{12}$$

Solve the equation t=\frac{-9±3\sqrt{19209}}{12} when ± is plus and when ± is minus.

$$t=\frac{\sqrt{19209}-3}{4} t=\frac{-\sqrt{19209}-3}{4}$$

Since a=t^{2}, the solutions are obtained by evaluating a=±\sqrt{t} for positive t.

$$a=\frac{\sqrt{\sqrt{19209}-3}}{2} a=-\frac{\sqrt{\sqrt{19209}-3}}{2}$$

Substitute \frac{\sqrt{\sqrt{19209}-3}}{2} for a in the equation \sqrt{\frac{3}{4}}a^{2}=3\times \frac{1}{2}\sqrt{400-\frac{a^{2}}{2}}.

$$\sqrt{\frac{3}{4}}\times \left(\frac{\sqrt{\sqrt{19209}-3}}{2}\right)^{2}=3\times \frac{1}{2}\sqrt{400-\frac{\left(\frac{\sqrt{\sqrt{19209}-3}}{2}\right)^{2}}{2}}$$

Simplify. The value a=\frac{\sqrt{\sqrt{19209}-3}}{2} satisfies the equation.

$$\frac{3}{8}\times 6403^{\frac{1}{2}}-\frac{3}{8}\times 3^{\frac{1}{2}}=\frac{3}{8}\times 6403^{\frac{1}{2}}-\frac{3}{8}\times 3^{\frac{1}{2}}$$

Substitute -\frac{\sqrt{\sqrt{19209}-3}}{2} for a in the equation \sqrt{\frac{3}{4}}a^{2}=3\times \frac{1}{2}\sqrt{400-\frac{a^{2}}{2}}.

$$\sqrt{\frac{3}{4}}\left(-\frac{\sqrt{\sqrt{19209}-3}}{2}\right)^{2}=3\times \frac{1}{2}\sqrt{400-\frac{\left(-\frac{\sqrt{\sqrt{19209}-3}}{2}\right)^{2}}{2}}$$

Simplify. The value a=-\frac{\sqrt{\sqrt{19209}-3}}{2} satisfies the equation.

$$\frac{3}{8}\times 6403^{\frac{1}{2}}-\frac{3}{8}\times 3^{\frac{1}{2}}=\frac{3}{8}\times 6403^{\frac{1}{2}}-\frac{3}{8}\times 3^{\frac{1}{2}}$$

List all solutions of \sqrt{\frac{3}{4}}a^{2}=\frac{3}{2}\sqrt{-\frac{a^{2}}{2}+400}.

$$a=\frac{\sqrt{\sqrt{19209}-3}}{2} a=-\frac{\sqrt{\sqrt{19209}-3}}{2}$$