Solve sqrt{4-x^2}=2-2/3left(3-xright)

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

Solve for x

$$x = \frac{6 \sqrt{13}}{13} \approx 1.664100589$$

Solution Steps

Square both sides of the equation.

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

Calculate \sqrt{4-x^{2}} to the power of 2 and get 4-x^{2}.

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

Use the distributive property to multiply -\frac{2}{3} by 3-x.

$$4-x^{2}=\left(2-2+\frac{2}{3}x\right)^{2}$$

Subtract 2 from 2 to get 0.

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

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

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

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

$$4-x^{2}=\frac{4}{9}x^{2}$$

Subtract \frac{4}{9}x^{2} from both sides.

$$4-x^{2}-\frac{4}{9}x^{2}=0$$

Combine -x^{2} and -\frac{4}{9}x^{2} to get -\frac{13}{9}x^{2}.

$$4-\frac{13}{9}x^{2}=0$$

Subtract 4 from both sides. Anything subtracted from zero gives its negation.

$$-\frac{13}{9}x^{2}=-4$$

Multiply both sides by -\frac{9}{13}, the reciprocal of -\frac{13}{9}.

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

Multiply -4 and -\frac{9}{13} to get \frac{36}{13}.

$$x^{2}=\frac{36}{13}$$

Take the square root of both sides of the equation.

$$x=\frac{6\sqrt{13}}{13} x=-\frac{6\sqrt{13}}{13}$$

Substitute \frac{6\sqrt{13}}{13} for x in the equation \sqrt{4-x^{2}}=2-\frac{2}{3}\left(3-x\right).

$$\sqrt{4-\left(\frac{6\sqrt{13}}{13}\right)^{2}}=2-\frac{2}{3}\left(3-\frac{6\sqrt{13}}{13}\right)$$

Simplify. The value x=\frac{6\sqrt{13}}{13} satisfies the equation.

$$\frac{4}{13}\times 13^{\frac{1}{2}}=\frac{4}{13}\times 13^{\frac{1}{2}}$$

Substitute -\frac{6\sqrt{13}}{13} for x in the equation \sqrt{4-x^{2}}=2-\frac{2}{3}\left(3-x\right).

$$\sqrt{4-\left(-\frac{6\sqrt{13}}{13}\right)^{2}}=2-\frac{2}{3}\left(3-\left(-\frac{6\sqrt{13}}{13}\right)\right)$$

Simplify. The value x=-\frac{6\sqrt{13}}{13} does not satisfy the equation because the left and the right hand side have opposite signs.

$$\frac{4}{13}\times 13^{\frac{1}{2}}=-\frac{4}{13}\times 13^{\frac{1}{2}}$$

Equation \sqrt{4-x^{2}}=-\frac{2\left(3-x\right)}{3}+2 has a unique solution.

$$x=\frac{6\sqrt{13}}{13}$$