Solve sqrt{-x-1}=x+3

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

Solve for x

$$x=-2$$

Solution Steps

Square both sides of the equation.

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

Calculate \sqrt{-x-1} to the power of 2 and get -x-1.

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

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

$$-x-1=x^{2}+6x+9$$

Subtract x^{2} from both sides.

$$-x-1-x^{2}=6x+9$$

Subtract 6x from both sides.

$$-x-1-x^{2}-6x=9$$

Subtract 9 from both sides.

$$-x-1-x^{2}-6x-9=0$$

Subtract 9 from -1 to get -10.

$$-x-10-x^{2}-6x=0$$

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

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

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 \sqrt{-x-1}=x+3.

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

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

$$1=1$$

Substitute -5 for x in the equation \sqrt{-x-1}=x+3.

$$\sqrt{-\left(-5\right)-1}=-5+3$$

Simplify. The value x=-5 does not satisfy the equation because the left and the right hand side have opposite signs.

$$2=-2$$

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

$$x=-2$$