Solve frac{sqrt{3}}{2}sqrt{1-x^2}+1/2x=frac{4sqrt{3}}{5}

$$\frac{ \sqrt{ 3 } }{ 2 } \sqrt{ 1- { x }^{ 2 } } + \frac{ 1 }{ 2 } x = \frac{ 4 \sqrt{ 3 } }{ 5 }$$

Solve for x (complex solution)

$$x=-\frac{\sqrt{69}i}{10}+\frac{2\sqrt{3}}{5}\approx 0.692820323-0.830662386i$$

Solution Steps

Subtract \frac{1}{2}x from both sides of the equation.

$$\frac{\sqrt{3}}{2}\sqrt{1-x^{2}}=\frac{4\sqrt{3}}{5}-\frac{1}{2}x$$

Multiply both sides of the equation by 10, the least common multiple of 2,5.

$$5\sqrt{3}\sqrt{1-x^{2}}=2\times 4\sqrt{3}-5x$$

Multiply 2 and 4 to get 8.

$$5\sqrt{3}\sqrt{1-x^{2}}=8\sqrt{3}-5x$$

Square both sides of the equation.

$$\left(5\sqrt{3}\sqrt{1-x^{2}}\right)^{2}=\left(8\sqrt{3}-5x\right)^{2}$$

Expand \left(5\sqrt{3}\sqrt{1-x^{2}}\right)^{2}.

$$5^{2}\left(\sqrt{3}\right)^{2}\left(\sqrt{1-x^{2}}\right)^{2}=\left(8\sqrt{3}-5x\right)^{2}$$

Calculate 5 to the power of 2 and get 25.

$$25\left(\sqrt{3}\right)^{2}\left(\sqrt{1-x^{2}}\right)^{2}=\left(8\sqrt{3}-5x\right)^{2}$$

The square of \sqrt{3} is 3.

$$25\times 3\left(\sqrt{1-x^{2}}\right)^{2}=\left(8\sqrt{3}-5x\right)^{2}$$

Multiply 25 and 3 to get 75.

$$75\left(\sqrt{1-x^{2}}\right)^{2}=\left(8\sqrt{3}-5x\right)^{2}$$

Calculate \sqrt{1-x^{2}} to the power of 2 and get 1-x^{2}.

$$75\left(1-x^{2}\right)=\left(8\sqrt{3}-5x\right)^{2}$$

Use the distributive property to multiply 75 by 1-x^{2}.

$$75-75x^{2}=\left(8\sqrt{3}-5x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(8\sqrt{3}-5x\right)^{2}.

$$75-75x^{2}=64\left(\sqrt{3}\right)^{2}-80\sqrt{3}x+25x^{2}$$

The square of \sqrt{3} is 3.

$$75-75x^{2}=64\times 3-80\sqrt{3}x+25x^{2}$$

Multiply 64 and 3 to get 192.

$$75-75x^{2}=192-80\sqrt{3}x+25x^{2}$$

Subtract 192 from both sides.

$$75-75x^{2}-192=-80\sqrt{3}x+25x^{2}$$

Subtract 192 from 75 to get -117.

$$-117-75x^{2}=-80\sqrt{3}x+25x^{2}$$

Add 80\sqrt{3}x to both sides.

$$-117-75x^{2}+80\sqrt{3}x=25x^{2}$$

Subtract 25x^{2} from both sides.

$$-117-75x^{2}+80\sqrt{3}x-25x^{2}=0$$

Combine -75x^{2} and -25x^{2} to get -100x^{2}.

$$-117-100x^{2}+80\sqrt{3}x=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-100x^{2}+80\sqrt{3}x-117=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -100 for a, 80\sqrt{3} for b, and -117 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-80\sqrt{3}±\sqrt{\left(80\sqrt{3}\right)^{2}-4\left(-100\right)\left(-117\right)}}{2\left(-100\right)}$$

Square 80\sqrt{3}.

$$x=\frac{-80\sqrt{3}±\sqrt{19200-4\left(-100\right)\left(-117\right)}}{2\left(-100\right)}$$

Multiply -4 times -100.

$$x=\frac{-80\sqrt{3}±\sqrt{19200+400\left(-117\right)}}{2\left(-100\right)}$$

Multiply 400 times -117.

$$x=\frac{-80\sqrt{3}±\sqrt{19200-46800}}{2\left(-100\right)}$$

Add 19200 to -46800.

$$x=\frac{-80\sqrt{3}±\sqrt{-27600}}{2\left(-100\right)}$$

Take the square root of -27600.

$$x=\frac{-80\sqrt{3}±20\sqrt{69}i}{2\left(-100\right)}$$

Multiply 2 times -100.

$$x=\frac{-80\sqrt{3}±20\sqrt{69}i}{-200}$$

Now solve the equation x=\frac{-80\sqrt{3}±20\sqrt{69}i}{-200} when ± is plus. Add -80\sqrt{3} to 20i\sqrt{69}.

$$x=\frac{-80\sqrt{3}+20\sqrt{69}i}{-200}$$

Divide -80\sqrt{3}+20i\sqrt{69} by -200.

$$x=-\frac{\sqrt{69}i}{10}+\frac{2\sqrt{3}}{5}$$

Now solve the equation x=\frac{-80\sqrt{3}±20\sqrt{69}i}{-200} when ± is minus. Subtract 20i\sqrt{69} from -80\sqrt{3}.

$$x=\frac{-20\sqrt{69}i-80\sqrt{3}}{-200}$$

Divide -80\sqrt{3}-20i\sqrt{69} by -200.

$$x=\frac{\sqrt{69}i}{10}+\frac{2\sqrt{3}}{5}$$

The equation is now solved.

$$x=-\frac{\sqrt{69}i}{10}+\frac{2\sqrt{3}}{5} x=\frac{\sqrt{69}i}{10}+\frac{2\sqrt{3}}{5}$$

Substitute -\frac{\sqrt{69}i}{10}+\frac{2\sqrt{3}}{5} for x in the equation \frac{\sqrt{3}}{2}\sqrt{1-x^{2}}+\frac{1}{2}x=\frac{4\sqrt{3}}{5}.

$$\frac{\sqrt{3}}{2}\sqrt{1-\left(-\frac{\sqrt{69}i}{10}+\frac{2\sqrt{3}}{5}\right)^{2}}+\frac{1}{2}\left(-\frac{\sqrt{69}i}{10}+\frac{2\sqrt{3}}{5}\right)=\frac{4\sqrt{3}}{5}$$

Simplify. The value x=-\frac{\sqrt{69}i}{10}+\frac{2\sqrt{3}}{5} satisfies the equation.

$$\frac{4}{5}\times 3^{\frac{1}{2}}=\frac{4}{5}\times 3^{\frac{1}{2}}$$

Substitute \frac{\sqrt{69}i}{10}+\frac{2\sqrt{3}}{5} for x in the equation \frac{\sqrt{3}}{2}\sqrt{1-x^{2}}+\frac{1}{2}x=\frac{4\sqrt{3}}{5}.

$$\frac{\sqrt{3}}{2}\sqrt{1-\left(\frac{\sqrt{69}i}{10}+\frac{2\sqrt{3}}{5}\right)^{2}}+\frac{1}{2}\left(\frac{\sqrt{69}i}{10}+\frac{2\sqrt{3}}{5}\right)=\frac{4\sqrt{3}}{5}$$

Simplify. The value x=\frac{\sqrt{69}i}{10}+\frac{2\sqrt{3}}{5} does not satisfy the equation.

$$-\frac{2}{5}\times 3^{\frac{1}{2}}+\frac{1}{10}i\times 69^{\frac{1}{2}}=\frac{4}{5}\times 3^{\frac{1}{2}}$$

Equation 5\sqrt{3}\sqrt{1-x^{2}}=-5x+8\sqrt{3} has a unique solution.

$$x=-\frac{\sqrt{69}i}{10}+\frac{2\sqrt{3}}{5}$$