Solve sqrt{1-x}+1=3x

$$\sqrt { 1 - x } + 1 = 3 x$$

Solve for x

$$x=\frac{5}{9}\approx 0.555555556$$

Solution Steps

Subtract 1 from both sides of the equation.

$$\sqrt{1-x}=3x-1$$

Square both sides of the equation.

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

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

$$1-x=\left(3x-1\right)^{2}$$

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

$$1-x=9x^{2}-6x+1$$

Subtract 9x^{2} from both sides.

$$1-x-9x^{2}=-6x+1$$

Add 6x to both sides.

$$1-x-9x^{2}+6x=1$$

Combine -x and 6x to get 5x.

$$1+5x-9x^{2}=1$$

Subtract 1 from both sides.

$$1+5x-9x^{2}-1=0$$

Subtract 1 from 1 to get 0.

$$5x-9x^{2}=0$$

Factor out x.

$$x\left(5-9x\right)=0$$

To find equation solutions, solve x=0 and 5-9x=0.

$$x=0 x=\frac{5}{9}$$

Substitute 0 for x in the equation \sqrt{1-x}+1=3x.

$$\sqrt{1-0}+1=3\times 0$$

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

$$2=0$$

Substitute \frac{5}{9} for x in the equation \sqrt{1-x}+1=3x.

$$\sqrt{1-\frac{5}{9}}+1=3\times \frac{5}{9}$$

Simplify. The value x=\frac{5}{9} satisfies the equation.

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

Equation \sqrt{1-x}=3x-1 has a unique solution.

$$x=\frac{5}{9}$$