Solve sqrt{4-6x-x^2}=x+4

$$\sqrt { 4 - 6 x - x ^ { 2 } } = x + 4$$

Solve for x

$$x=-1$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{4-6x-x^{2}}\right)^{2}=\left(x+4\right)^{2}$$

Calculate \sqrt{4-6x-x^{2}} to the power of 2 and get 4-6x-x^{2}.

$$4-6x-x^{2}=\left(x+4\right)^{2}$$

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

$$4-6x-x^{2}=x^{2}+8x+16$$

Subtract x^{2} from both sides.

$$4-6x-x^{2}-x^{2}=8x+16$$

Combine -x^{2} and -x^{2} to get -2x^{2}.

$$4-6x-2x^{2}=8x+16$$

Subtract 8x from both sides.

$$4-6x-2x^{2}-8x=16$$

Combine -6x and -8x to get -14x.

$$4-14x-2x^{2}=16$$

Subtract 16 from both sides.

$$4-14x-2x^{2}-16=0$$

Subtract 16 from 4 to get -12.

$$-12-14x-2x^{2}=0$$

Divide both sides by 2.

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

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{4-6x-x^{2}}=x+4.

$$\sqrt{4-6\left(-1\right)-\left(-1\right)^{2}}=-1+4$$

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

$$3=3$$

Substitute -6 for x in the equation \sqrt{4-6x-x^{2}}=x+4.

$$\sqrt{4-6\left(-6\right)-\left(-6\right)^{2}}=-6+4$$

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

$$2=-2$$

Equation \sqrt{4-6x-x^{2}}=x+4 has a unique solution.

$$x=-1$$