Solve sqrt{3-x}-x=3

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

Solve for x

$$x=-1$$

Solution Steps

Subtract -x from both sides of the equation.

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

Square both sides of the equation.

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

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

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

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

$$3-x=9+6x+x^{2}$$

Subtract 9 from both sides.

$$3-x-9=6x+x^{2}$$

Subtract 9 from 3 to get -6.

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

Subtract 6x from both sides.

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

Combine -x and -6x to get -7x.

$$-6-7x=x^{2}$$

Subtract x^{2} from both sides.

$$-6-7x-x^{2}=0$$

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

$$-x^{2}-7x-6=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-6. To find a and b, set up a system to be solved.

$$a+b=-7 ab=-\left(-6\right)=6$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 6.

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

Calculate the sum for each pair.

$$-1-6=-7 -2-3=-5$$

The solution is the pair that gives sum -7.

$$a=-1 b=-6$$

Rewrite -x^{2}-7x-6 as \left(-x^{2}-x\right)+\left(-6x-6\right).

$$\left(-x^{2}-x\right)+\left(-6x-6\right)$$

Factor out x in the first and 6 in the second group.

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

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

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

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

$$x=-1 x=-6$$

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

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

Simplify. The value x=-1 satisfies the equation.

$$3=3$$

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

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

Simplify. The value x=-6 does not satisfy the equation.

$$9=3$$

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

$$x=-1$$