Solve 2x-1=-sqrt{2-x}

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

Solve for x

$$x=-\frac{1}{4}=-0.25$$

Solution Steps

Add \sqrt{2-x} to both sides.

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

Add 1 to both sides. Anything plus zero gives itself.

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

Subtract 2x from both sides of the equation.

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

Square both sides of the equation.

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

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

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

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

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

Add 4x to both sides.

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

Combine -x and 4x to get 3x.

$$2+3x=1+4x^{2}$$

Subtract 4x^{2} from both sides.

$$2+3x-4x^{2}=1$$

Subtract 1 from both sides.

$$2+3x-4x^{2}-1=0$$

Subtract 1 from 2 to get 1.

$$1+3x-4x^{2}=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-4x^{2}+3x+1=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -4x^{2}+ax+bx+1. To find a and b, set up a system to be solved.

$$a+b=3 ab=-4=-4$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -4.

$$-1,4 -2,2$$

Calculate the sum for each pair.

$$-1+4=3 -2+2=0$$

The solution is the pair that gives sum 3.

$$a=4 b=-1$$

Rewrite -4x^{2}+3x+1 as \left(-4x^{2}+4x\right)+\left(-x+1\right).

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

Factor out 4x in -4x^{2}+4x.

$$4x\left(-x+1\right)-x+1$$

Factor out common term -x+1 by using distributive property.

$$\left(-x+1\right)\left(4x+1\right)$$

To find equation solutions, solve -x+1=0 and 4x+1=0.

$$x=1 x=-\frac{1}{4}$$

Substitute 1 for x in the equation 2x-1=-\sqrt{2-x}.

$$2\times 1-1=-\sqrt{2-1}$$

Simplify. The value x=1 does not satisfy the equation because the left and the right hand side have opposite signs.

$$1=-1$$

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

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

Simplify. The value x=-\frac{1}{4} satisfies the equation.

$$-\frac{3}{2}=-\frac{3}{2}$$

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

$$x=-\frac{1}{4}$$