Solve 2-x=3-sqrt{7-3x}

$$2 - x = 3 - \sqrt { 7 - 3 x }$$

Solve for x

$$x=1$$

Solution Steps

Subtract 3 from both sides of the equation.

$$2-x-3=-\sqrt{7-3x}$$

Subtract 3 from 2 to get -1.

$$-1-x=-\sqrt{7-3x}$$

Square both sides of the equation.

$$\left(-1-x\right)^{2}=\left(-\sqrt{7-3x}\right)^{2}$$

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

$$1+2x+x^{2}=\left(-\sqrt{7-3x}\right)^{2}$$

Expand \left(-\sqrt{7-3x}\right)^{2}.

$$1+2x+x^{2}=\left(-1\right)^{2}\left(\sqrt{7-3x}\right)^{2}$$

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

$$1+2x+x^{2}=1\left(\sqrt{7-3x}\right)^{2}$$

Calculate \sqrt{7-3x} to the power of 2 and get 7-3x.

$$1+2x+x^{2}=1\left(7-3x\right)$$

Use the distributive property to multiply 1 by 7-3x.

$$1+2x+x^{2}=7-3x$$

Subtract 7 from both sides.

$$1+2x+x^{2}-7=-3x$$

Subtract 7 from 1 to get -6.

$$-6+2x+x^{2}=-3x$$

Add 3x to both sides.

$$-6+2x+x^{2}+3x=0$$

Combine 2x and 3x to get 5x.

$$-6+5x+x^{2}=0$$

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

$$x^{2}+5x-6=0$$

To solve the equation, factor x^{2}+5x-6 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.

$$a+b=5 ab=-6$$

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 -6.

$$-1,6 -2,3$$

Calculate the sum for each pair.

$$-1+6=5 -2+3=1$$

The solution is the pair that gives sum 5.

$$a=-1 b=6$$

Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.

$$\left(x-1\right)\left(x+6\right)$$

To find equation solutions, solve x-1=0 and x+6=0.

$$x=1 x=-6$$

Substitute 1 for x in the equation 2-x=3-\sqrt{7-3x}.

$$2-1=3-\sqrt{7-3}$$

Simplify. The value x=1 satisfies the equation.

$$1=1$$

Substitute -6 for x in the equation 2-x=3-\sqrt{7-3x}.

$$2-\left(-6\right)=3-\sqrt{7-3\left(-6\right)}$$

Simplify. The value x=-6 does not satisfy the equation because the left and the right hand side have opposite signs.

$$8=-2$$

Equation -x-1=-\sqrt{7-3x} has a unique solution.

$$x=1$$