Solve 12/13sqrt{1-x^2}-5/13x=3/5

$$\frac { 12 } { 13 } \sqrt { 1 - x ^ { 2 } } - \frac { 5 } { 13 } x = \frac { 3 } { 5 }$$

Solve for x

$$x=\frac{33}{65}\approx 0.507692308$$

$$x=-\frac{63}{65}\approx -0.969230769$$

Solution Steps

Subtract -\frac{5}{13}x from both sides of the equation.

$$\frac{12}{13}\sqrt{1-x^{2}}=\frac{3}{5}+\frac{5}{13}x$$

Square both sides of the equation.

$$\left(\frac{12}{13}\sqrt{1-x^{2}}\right)^{2}=\left(\frac{3}{5}+\frac{5}{13}x\right)^{2}$$

Expand \left(\frac{12}{13}\sqrt{1-x^{2}}\right)^{2}.

$$\left(\frac{12}{13}\right)^{2}\left(\sqrt{1-x^{2}}\right)^{2}=\left(\frac{3}{5}+\frac{5}{13}x\right)^{2}$$

Calculate \frac{12}{13} to the power of 2 and get \frac{144}{169}.

$$\frac{144}{169}\left(\sqrt{1-x^{2}}\right)^{2}=\left(\frac{3}{5}+\frac{5}{13}x\right)^{2}$$

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

$$\frac{144}{169}\left(1-x^{2}\right)=\left(\frac{3}{5}+\frac{5}{13}x\right)^{2}$$

Use the distributive property to multiply \frac{144}{169} by 1-x^{2}.

$$\frac{144}{169}-\frac{144}{169}x^{2}=\left(\frac{3}{5}+\frac{5}{13}x\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\frac{3}{5}+\frac{5}{13}x\right)^{2}.

$$\frac{144}{169}-\frac{144}{169}x^{2}=\frac{9}{25}+\frac{6}{13}x+\frac{25}{169}x^{2}$$

Subtract \frac{9}{25} from both sides.

$$\frac{144}{169}-\frac{144}{169}x^{2}-\frac{9}{25}=\frac{6}{13}x+\frac{25}{169}x^{2}$$

Subtract \frac{9}{25} from \frac{144}{169} to get \frac{2079}{4225}.

$$\frac{2079}{4225}-\frac{144}{169}x^{2}=\frac{6}{13}x+\frac{25}{169}x^{2}$$

Subtract \frac{6}{13}x from both sides.

$$\frac{2079}{4225}-\frac{144}{169}x^{2}-\frac{6}{13}x=\frac{25}{169}x^{2}$$

Subtract \frac{25}{169}x^{2} from both sides.

$$\frac{2079}{4225}-\frac{144}{169}x^{2}-\frac{6}{13}x-\frac{25}{169}x^{2}=0$$

Combine -\frac{144}{169}x^{2} and -\frac{25}{169}x^{2} to get -x^{2}.

$$\frac{2079}{4225}-x^{2}-\frac{6}{13}x=0$$

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.

$$-x^{2}-\frac{6}{13}x+\frac{2079}{4225}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -\frac{6}{13} for b, and \frac{2079}{4225} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\frac{6}{13}\right)±\sqrt{\left(-\frac{6}{13}\right)^{2}-4\left(-1\right)\times \frac{2079}{4225}}}{2\left(-1\right)}$$

Square -\frac{6}{13} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-\frac{6}{13}\right)±\sqrt{\frac{36}{169}-4\left(-1\right)\times \frac{2079}{4225}}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-\frac{6}{13}\right)±\sqrt{\frac{36}{169}+4\times \frac{2079}{4225}}}{2\left(-1\right)}$$

Multiply 4 times \frac{2079}{4225}.

$$x=\frac{-\left(-\frac{6}{13}\right)±\sqrt{\frac{36}{169}+\frac{8316}{4225}}}{2\left(-1\right)}$$

Add \frac{36}{169} to \frac{8316}{4225} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-\frac{6}{13}\right)±\sqrt{\frac{9216}{4225}}}{2\left(-1\right)}$$

Take the square root of \frac{9216}{4225}.

$$x=\frac{-\left(-\frac{6}{13}\right)±\frac{96}{65}}{2\left(-1\right)}$$

The opposite of -\frac{6}{13} is \frac{6}{13}.

$$x=\frac{\frac{6}{13}±\frac{96}{65}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{\frac{6}{13}±\frac{96}{65}}{-2}$$

Now solve the equation x=\frac{\frac{6}{13}±\frac{96}{65}}{-2} when ± is plus. Add \frac{6}{13} to \frac{96}{65} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{\frac{126}{65}}{-2}$$

Divide \frac{126}{65} by -2.

$$x=-\frac{63}{65}$$

Now solve the equation x=\frac{\frac{6}{13}±\frac{96}{65}}{-2} when ± is minus. Subtract \frac{96}{65} from \frac{6}{13} by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{\frac{66}{65}}{-2}$$

Divide -\frac{66}{65} by -2.

$$x=\frac{33}{65}$$

The equation is now solved.

$$x=-\frac{63}{65} x=\frac{33}{65}$$

Substitute -\frac{63}{65} for x in the equation \frac{12}{13}\sqrt{1-x^{2}}-\frac{5}{13}x=\frac{3}{5}.

$$\frac{12}{13}\sqrt{1-\left(-\frac{63}{65}\right)^{2}}-\frac{5}{13}\left(-\frac{63}{65}\right)=\frac{3}{5}$$

Simplify. The value x=-\frac{63}{65} satisfies the equation.

$$\frac{3}{5}=\frac{3}{5}$$

Substitute \frac{33}{65} for x in the equation \frac{12}{13}\sqrt{1-x^{2}}-\frac{5}{13}x=\frac{3}{5}.

$$\frac{12}{13}\sqrt{1-\left(\frac{33}{65}\right)^{2}}-\frac{5}{13}\times \frac{33}{65}=\frac{3}{5}$$

Simplify. The value x=\frac{33}{65} satisfies the equation.

$$\frac{3}{5}=\frac{3}{5}$$

List all solutions of \frac{12\sqrt{1-x^{2}}}{13}=\frac{5x}{13}+\frac{3}{5}.

$$x=-\frac{63}{65} x=\frac{33}{65}$$