Solve 4x=1+sqrt{1-x^2}

$$4 x = 1 + \sqrt { 1 - x ^ { 2 } }$$

Solve for x

$$x=\frac{8}{17}\approx 0.470588235$$

Solution Steps

Subtract 1 from both sides of the equation.

$$4x-1=\sqrt{1-x^{2}}$$

Square both sides of the equation.

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4x-1\right)^{2}.

$$16x^{2}-8x+1=\left(\sqrt{1-x^{2}}\right)^{2}$$

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

$$16x^{2}-8x+1=1-x^{2}$$

Subtract 1 from both sides.

$$16x^{2}-8x+1-1=-x^{2}$$

Subtract 1 from 1 to get 0.

$$16x^{2}-8x=-x^{2}$$

Add x^{2} to both sides.

$$16x^{2}-8x+x^{2}=0$$

Combine 16x^{2} and x^{2} to get 17x^{2}.

$$17x^{2}-8x=0$$

Factor out x.

$$x\left(17x-8\right)=0$$

To find equation solutions, solve x=0 and 17x-8=0.

$$x=0 x=\frac{8}{17}$$

Substitute 0 for x in the equation 4x=1+\sqrt{1-x^{2}}.

$$4\times 0=1+\sqrt{1-0^{2}}$$

Simplify. The value x=0 does not satisfy the equation.

$$0=2$$

Substitute \frac{8}{17} for x in the equation 4x=1+\sqrt{1-x^{2}}.

$$4\times \frac{8}{17}=1+\sqrt{1-\left(\frac{8}{17}\right)^{2}}$$

Simplify. The value x=\frac{8}{17} satisfies the equation.

$$\frac{32}{17}=\frac{32}{17}$$

Equation 4x-1=\sqrt{1-x^{2}} has a unique solution.

$$x=\frac{8}{17}$$