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

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

Solve for x

$$x=-2$$

Solution Steps

Subtract 4^{1} from both sides of the equation.

$$-\sqrt{x+6}=-x-4^{1}$$

Calculate 4 to the power of 1 and get 4.

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

Square both sides of the equation.

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

Expand \left(-\sqrt{x+6}\right)^{2}.

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

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

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

Calculate \sqrt{x+6} to the power of 2 and get x+6.

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

Use the distributive property to multiply 1 by x+6.

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

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

$$x+6=\left(-x\right)^{2}-8\left(-x\right)+16$$

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

$$x+6=x^{2}-8\left(-x\right)+16$$

Multiply -8 and -1 to get 8.

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

Subtract x^{2} from both sides.

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

Subtract 8x from both sides.

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

Combine x and -8x to get -7x.

$$-7x+6-x^{2}=16$$

Subtract 16 from both sides.

$$-7x+6-x^{2}-16=0$$

Subtract 16 from 6 to get -10.

$$-7x-10-x^{2}=0$$

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

$$-x^{2}-7x-10=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-10. To find a and b, set up a system to be solved.

$$a+b=-7 ab=-\left(-10\right)=10$$

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 10.

$$-1,-10 -2,-5$$

Calculate the sum for each pair.

$$-1-10=-11 -2-5=-7$$

The solution is the pair that gives sum -7.

$$a=-2 b=-5$$

Rewrite -x^{2}-7x-10 as \left(-x^{2}-2x\right)+\left(-5x-10\right).

$$\left(-x^{2}-2x\right)+\left(-5x-10\right)$$

Factor out x in the first and 5 in the second group.

$$x\left(-x-2\right)+5\left(-x-2\right)$$

Factor out common term -x-2 by using distributive property.

$$\left(-x-2\right)\left(x+5\right)$$

To find equation solutions, solve -x-2=0 and x+5=0.

$$x=-2 x=-5$$

Substitute -2 for x in the equation 4^{1}-\sqrt{x+6}=-x.

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

Simplify. The value x=-2 satisfies the equation.

$$2=2$$

Substitute -5 for x in the equation 4^{1}-\sqrt{x+6}=-x.

$$4^{1}-\sqrt{-5+6}=-\left(-5\right)$$

Simplify. The value x=-5 does not satisfy the equation.

$$3=5$$

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

$$x=-2$$