Solve sqrt{9x^2-sqrt{36x^2-11}}=3x-1

$$\sqrt { 9 x ^ { 2 } - \sqrt { 36 x ^ { 2 } - 11 } } = 3 x - 1$$

Solve for x

$$x=1$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{9x^{2}-\sqrt{36x^{2}-11}}\right)^{2}=\left(3x-1\right)^{2}$$

Calculate \sqrt{9x^{2}-\sqrt{36x^{2}-11}} to the power of 2 and get 9x^{2}-\sqrt{36x^{2}-11}.

$$9x^{2}-\sqrt{36x^{2}-11}=\left(3x-1\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(3x-1\right)^{2}.

$$9x^{2}-\sqrt{36x^{2}-11}=9x^{2}-6x+1$$

Subtract 9x^{2} from both sides of the equation.

$$-\sqrt{36x^{2}-11}=9x^{2}-6x+1-9x^{2}$$

Combine 9x^{2} and -9x^{2} to get 0.

$$-\sqrt{36x^{2}-11}=-6x+1$$

Square both sides of the equation.

$$\left(-\sqrt{36x^{2}-11}\right)^{2}=\left(-6x+1\right)^{2}$$

Expand \left(-\sqrt{36x^{2}-11}\right)^{2}.

$$\left(-1\right)^{2}\left(\sqrt{36x^{2}-11}\right)^{2}=\left(-6x+1\right)^{2}$$

Calculate -1 to the power of 2 and get 1.

$$1\left(\sqrt{36x^{2}-11}\right)^{2}=\left(-6x+1\right)^{2}$$

Calculate \sqrt{36x^{2}-11} to the power of 2 and get 36x^{2}-11.

$$1\left(36x^{2}-11\right)=\left(-6x+1\right)^{2}$$

Use the distributive property to multiply 1 by 36x^{2}-11.

$$36x^{2}-11=\left(-6x+1\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(-6x+1\right)^{2}.

$$36x^{2}-11=36x^{2}-12x+1$$

Subtract 36x^{2} from both sides.

$$36x^{2}-11-36x^{2}=-12x+1$$

Combine 36x^{2} and -36x^{2} to get 0.

$$-11=-12x+1$$

Swap sides so that all variable terms are on the left hand side.

$$-12x+1=-11$$

Subtract 1 from both sides.

$$-12x=-11-1$$

Subtract 1 from -11 to get -12.

$$-12x=-12$$

Divide both sides by -12.

$$x=\frac{-12}{-12}$$

Divide -12 by -12 to get 1.

$$x=1$$

Substitute 1 for x in the equation \sqrt{9x^{2}-\sqrt{36x^{2}-11}}=3x-1.

$$\sqrt{9\times 1^{2}-\sqrt{36\times 1^{2}-11}}=3\times 1-1$$

Simplify. The value x=1 satisfies the equation.

$$2=2$$

Equation \sqrt{9x^{2}-\sqrt{36x^{2}-11}}=3x-1 has a unique solution.

$$x=1$$