Solve 4/5x-3/5sqrt{1-x^2}=5/13

$$\frac { 4 } { 5 } x - \frac { 3 } { 5 } \sqrt { 1 - x ^ { 2 } } = \frac { 5 } { 13 }$$

Solve for x

$$x=\frac{56}{65}\approx 0.861538462$$

Solution Steps

Subtract \frac{4}{5}x from both sides of the equation.

$$-\frac{3}{5}\sqrt{1-x^{2}}=\frac{5}{13}-\frac{4}{5}x$$

Square both sides of the equation.

$$\left(-\frac{3}{5}\sqrt{1-x^{2}}\right)^{2}=\left(\frac{5}{13}-\frac{4}{5}x\right)^{2}$$

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

$$\left(-\frac{3}{5}\right)^{2}\left(\sqrt{1-x^{2}}\right)^{2}=\left(\frac{5}{13}-\frac{4}{5}x\right)^{2}$$

Calculate -\frac{3}{5} to the power of 2 and get \frac{9}{25}.

$$\frac{9}{25}\left(\sqrt{1-x^{2}}\right)^{2}=\left(\frac{5}{13}-\frac{4}{5}x\right)^{2}$$

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

$$\frac{9}{25}\left(1-x^{2}\right)=\left(\frac{5}{13}-\frac{4}{5}x\right)^{2}$$

Use the distributive property to multiply \frac{9}{25} by 1-x^{2}.

$$\frac{9}{25}-\frac{9}{25}x^{2}=\left(\frac{5}{13}-\frac{4}{5}x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\frac{5}{13}-\frac{4}{5}x\right)^{2}.

$$\frac{9}{25}-\frac{9}{25}x^{2}=\frac{25}{169}-\frac{8}{13}x+\frac{16}{25}x^{2}$$

Subtract \frac{25}{169} from both sides.

$$\frac{9}{25}-\frac{9}{25}x^{2}-\frac{25}{169}=-\frac{8}{13}x+\frac{16}{25}x^{2}$$

Subtract \frac{25}{169} from \frac{9}{25} to get \frac{896}{4225}.

$$\frac{896}{4225}-\frac{9}{25}x^{2}=-\frac{8}{13}x+\frac{16}{25}x^{2}$$

Add \frac{8}{13}x to both sides.

$$\frac{896}{4225}-\frac{9}{25}x^{2}+\frac{8}{13}x=\frac{16}{25}x^{2}$$

Subtract \frac{16}{25}x^{2} from both sides.

$$\frac{896}{4225}-\frac{9}{25}x^{2}+\frac{8}{13}x-\frac{16}{25}x^{2}=0$$

Combine -\frac{9}{25}x^{2} and -\frac{16}{25}x^{2} to get -x^{2}.

$$\frac{896}{4225}-x^{2}+\frac{8}{13}x=0$$

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.

$$-x^{2}+\frac{8}{13}x+\frac{896}{4225}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, \frac{8}{13} for b, and \frac{896}{4225} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\frac{8}{13}±\sqrt{\left(\frac{8}{13}\right)^{2}-4\left(-1\right)\times \frac{896}{4225}}}{2\left(-1\right)}$$

Square \frac{8}{13} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\frac{8}{13}±\sqrt{\frac{64}{169}-4\left(-1\right)\times \frac{896}{4225}}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\frac{8}{13}±\sqrt{\frac{64}{169}+4\times \frac{896}{4225}}}{2\left(-1\right)}$$

Multiply 4 times \frac{896}{4225}.

$$x=\frac{-\frac{8}{13}±\sqrt{\frac{64}{169}+\frac{3584}{4225}}}{2\left(-1\right)}$$

Add \frac{64}{169} to \frac{3584}{4225} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\frac{8}{13}±\sqrt{\frac{5184}{4225}}}{2\left(-1\right)}$$

Take the square root of \frac{5184}{4225}.

$$x=\frac{-\frac{8}{13}±\frac{72}{65}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-\frac{8}{13}±\frac{72}{65}}{-2}$$

Now solve the equation x=\frac{-\frac{8}{13}±\frac{72}{65}}{-2} when ± is plus. Add -\frac{8}{13} to \frac{72}{65} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{\frac{32}{65}}{-2}$$

Divide \frac{32}{65} by -2.

$$x=-\frac{16}{65}$$

Now solve the equation x=\frac{-\frac{8}{13}±\frac{72}{65}}{-2} when ± is minus. Subtract \frac{72}{65} from -\frac{8}{13} by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{\frac{112}{65}}{-2}$$

Divide -\frac{112}{65} by -2.

$$x=\frac{56}{65}$$

The equation is now solved.

$$x=-\frac{16}{65} x=\frac{56}{65}$$

Substitute -\frac{16}{65} for x in the equation \frac{4}{5}x-\frac{3}{5}\sqrt{1-x^{2}}=\frac{5}{13}.

$$\frac{4}{5}\left(-\frac{16}{65}\right)-\frac{3}{5}\sqrt{1-\left(-\frac{16}{65}\right)^{2}}=\frac{5}{13}$$

Simplify. The value x=-\frac{16}{65} does not satisfy the equation because the left and the right hand side have opposite signs.

$$-\frac{253}{325}=\frac{5}{13}$$

Substitute \frac{56}{65} for x in the equation \frac{4}{5}x-\frac{3}{5}\sqrt{1-x^{2}}=\frac{5}{13}.

$$\frac{4}{5}\times \frac{56}{65}-\frac{3}{5}\sqrt{1-\left(\frac{56}{65}\right)^{2}}=\frac{5}{13}$$

Simplify. The value x=\frac{56}{65} satisfies the equation.

$$\frac{5}{13}=\frac{5}{13}$$

Equation -\frac{3\sqrt{1-x^{2}}}{5}=-\frac{4x}{5}+\frac{5}{13} has a unique solution.

$$x=\frac{56}{65}$$