Solve 18x=0x+36sqrt{1-x^2}

$$18 x = 0 x + 36 \sqrt { 1 - x ^ { 2 } }$$

Solve for x

$$x=\frac{2\sqrt{5}}{5}\approx 0.894427191$$

Solution Steps

Subtract 0 from both sides of the equation.

$$18x=36\sqrt{1-x^{2}}$$

Anything times zero gives zero.

$$18x+0=36\sqrt{1-x^{2}}$$

Anything plus zero gives itself.

$$18x=36\sqrt{1-x^{2}}$$

Square both sides of the equation.

$$\left(18x\right)^{2}=\left(36\sqrt{1-x^{2}}\right)^{2}$$

Expand \left(18x\right)^{2}.

$$18^{2}x^{2}=\left(36\sqrt{1-x^{2}}\right)^{2}$$

Calculate 18 to the power of 2 and get 324.

$$324x^{2}=\left(36\sqrt{1-x^{2}}\right)^{2}$$

Expand \left(36\sqrt{1-x^{2}}\right)^{2}.

$$324x^{2}=36^{2}\left(\sqrt{1-x^{2}}\right)^{2}$$

Calculate 36 to the power of 2 and get 1296.

$$324x^{2}=1296\left(\sqrt{1-x^{2}}\right)^{2}$$

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

$$324x^{2}=1296\left(1-x^{2}\right)$$

Use the distributive property to multiply 1296 by 1-x^{2}.

$$324x^{2}=1296-1296x^{2}$$

Add 1296x^{2} to both sides.

$$324x^{2}+1296x^{2}=1296$$

Combine 324x^{2} and 1296x^{2} to get 1620x^{2}.

$$1620x^{2}=1296$$

Divide both sides by 1620.

$$x^{2}=\frac{1296}{1620}$$

Reduce the fraction \frac{1296}{1620} to lowest terms by extracting and canceling out 324.

$$x^{2}=\frac{4}{5}$$

Take the square root of both sides of the equation.

$$x=\frac{2\sqrt{5}}{5} x=-\frac{2\sqrt{5}}{5}$$

Substitute \frac{2\sqrt{5}}{5} for x in the equation 18x=0x+36\sqrt{1-x^{2}}.

$$18\times \frac{2\sqrt{5}}{5}=0\times \frac{2\sqrt{5}}{5}+36\sqrt{1-\left(\frac{2\sqrt{5}}{5}\right)^{2}}$$

Simplify. The value x=\frac{2\sqrt{5}}{5} satisfies the equation.

$$\frac{36}{5}\times 5^{\frac{1}{2}}=\frac{36}{5}\times 5^{\frac{1}{2}}$$

Substitute -\frac{2\sqrt{5}}{5} for x in the equation 18x=0x+36\sqrt{1-x^{2}}.

$$18\left(-\frac{2\sqrt{5}}{5}\right)=0\left(-\frac{2\sqrt{5}}{5}\right)+36\sqrt{1-\left(-\frac{2\sqrt{5}}{5}\right)^{2}}$$

Simplify. The value x=-\frac{2\sqrt{5}}{5} does not satisfy the equation because the left and the right hand side have opposite signs.

$$-\frac{36}{5}\times 5^{\frac{1}{2}}=\frac{36}{5}\times 5^{\frac{1}{2}}$$

Equation 18x=36\sqrt{1-x^{2}} has a unique solution.

$$x=\frac{2\sqrt{5}}{5}$$